Parmenides introduces an early version of the problem of negative existentials. In modern times, this has been construed as a problem about the relationship between reference and meaning and the linguistic mechanics of singular terms, existential quantification, de re/de dicto distinctions, etc. For Parminedes, of course, the problem appears as a problem about the relation between thought and being. With that said, we can reconstruct Parmenides' basic reasoning a little bit anachronistically for the sake of illumination.
Consider the following inconsistent triad:
A sentence can only be true or false if it is meaningful.
E.g., 'Jack greens deskly' cannot be assigned a truth-value because it is syntactically ill-formed and cannot designate anything.
All parts of a sentence must be meaningful if the whole sentence is meaningful.
E.g., 'Jack runs deskly' has one meaningful part ('Jack runs') but is not meaningful as a whole and therefore cannot be assigned a truth-value.
If a singular term is meaningful, it must be about something.
The problem is that these three assumptions taken separately each seem to be rather plausible constraints. It is surprising then, that, taken together, we get an inconsistent theory of meaning.
In short, our innocuous assumptions make claims the form 'x does not exist' self-undermining. Consider your target statement (with the indexicals taken out to simplify): 'A bank account in John Doe's name does not exist.' In order for this statement to be possibly true, it must be meaningful (i.e., it must be evaluable for truth or falsity). But, by our assumption, in order for a statement to be meaningful it must be about something. But look at the content of our target statement: it says precisely that its subject-matter does not exist. Hence, if the statement is true, then it is about nothing and is therefore not meaningful. In sum, if we apply our three innocuous principles, we end up with a statement such that if it is true, then it cannot possibly be true or false.
Put simply, the result is that there are no true statements of the form 'x does not exist' and therefore they cannot be included in a scientific theory. For Parmenides, this came out as a kind of mutual subservience of thought and being. Backpedaling on the anachronisticity of the explanation of I've just given, we must note that Parmenides of course did not have the notion of 'assigning truth-values to propositions', forget about the definition of logical terms as truth-functions. The idea that propositions are the bearers of truth-values is a logical innovation in the Stoics long after Parmenides' influence is felt in Greek philosophy. Plausibly, whatever the truth-value bearers were for Parmiendes, they were the sort of thing that could not be 'false' because they had to have existence in thought. For example, Epicureans held the bearers of truth-values to be multiplicities of atoms acting on the sense-organs. In this sense, there are no entities like the propositions of today's philosophers that can be true or false---everything that is is, and nothing else besides. For example, a rock exists. It would be absurd to say that "the rock is false." Pre-Stoic philosophers did not have a conception of entities that could be true or false. For us, the idea of propositions being truth-bearers is built into the way we learn logic and critical thinking. But the pre-Socratics had not pried apart existence and truth. That would be the result of logical innovations to which Parmenides, Heraclitus, Epicurus, and so on were not privileged.
Now, you are unhappy with the conclusion you've reached by applying Parmenides' view. But your objection to the conclusion is no objection to Parmenides. First, you say:
If I go to the bank and give my name, nothing will happen no matter how much I think about there not existing a 'bank account in my name.'
Yes, and this is no problem for Parmenides. Consider, just because Parmenides has said you cannot cognize the statement 'A bank account in John Doe's name does not exist' does not at all imply that we should expect the world to behave as if the negation of this statement were true. Just because we cannot assign a truth-value to the statement 'A bank account in John Doe's name does not exist' does not mean that we should expect the world to evince its opposite. Actually, that's just the problem: statements of the form of negative existentials fail to make an ontologically classificatory distinction at all by virtue of which we could compare the world as it is to the world as it is signified in the proposition. It's like if I asked you what color a color wheel was, when the color wheel is the very standard for assigning colors in the first place; or if I asked you how long a meter stick was, etc. If you started measuring a meter stick with itself, it wouldn't be out of place for someone to tell you you're not doing what you thought you were doing. Remember, Parmenides is saying that 'nothingness' is off limits for inclusion in scientific theorizing. Hence, what has happened in this example is simply that you took yourself to be doing something that you weren't actually doing: thinking.
This leads to the second weakness of your objection, which is essentially that you can't accept Parmenides' anti-intuitive result. But Parmenides' account results in a highly non-intuitive monism that dispenses with the testimony of the senses. For example, we perceive the world around is as being in motion, but notice that claims about motion require claims about non-existents, since motion is precisely the passing of states of affairs into and out of existence. The famous rejoinder to Parmenides' denial of the reality of motion is to stand up and walk across the room, demonstrating motion. But of course the claim that Parmenides' conclusion contradicts common sense is no serious objection to Parmenides, since the metaphilosophical claim Parmenides is making is that where science and sense-perception diverge, science is to be trusted. For example, the problem of infinity in the analysis of motion suggests that any measurement of motion in the world as given by sense-perception will be intolerably noisy. Hence, we have no scientific account of motion since it cannot be measured and quantified (this was precisely Zeno's paradox, and you'll notice reiterations of the measurement problem in quantum physics, etc.).
In general, the pre-Socratics were concerned with problems they encountered early on with applying mathematical measurements to the world as encountered by sense-perception. Parmenides takes a hard-line approach towards favoring science and declares the sensory world to be an illusion wherever mathematical measurements are noisy. On the other side, we have Heraclitus, whose views challenge the notion of cognition by declaring the world to be a flux that is so unstable that our cognition cannot grab hold of it. Against this background (along with the socio-political turmoil of Athens at the time of the Peloponnesian war), Plato attempts to justify science by holding that our cognition grasps on to stable Forms (counteracting Heraclitus' skepticism) and that the sensory world can be said to resemble those forms more or less perfectly (hence recovering from Parmenides the resemblance of the world as revealed by scientific cognition and the world as revealed by the senses). Indeed, science in some important sense depends on the resemblance of the lower forms to the sensory realms, since it is these resemblances that will enable our minds to eventually move freely in philosophical thought about the highest Forms such as beauty, truth, and goodness. Contrast with Aristotle, who decisively quelled the pre-Socratic debate about the relationship between measurement and sense-perception with his theory of abstraction, on which nothing reaches our cognition without first entering into our sense-experience. This Aristotlean innovation is low key in the background of why we moderns tend to think of science as matching up with the world of the senses as being rather unproblematic. But this problem plagued the pre-Socratics and was the impetus behind Plato's theory of the Forms.