Occam's razor states that, everything else being equal, the theory with the least number of assumptions is more likely to be true. This has been formalized as Solomonoff's theory of inductive inference.
Heuristically it seems reasonable that given two theories, A and B, if A is more parsimonious than B, then A is harder to falsify: the theory that is more likely to be true is less likely to be falsified.
But I do not feel comfortable with just a simple heuristic argument. I am looking for better justification or an argument as to why it is not the case.
Here are some more thoughts, since I do not want to answer my own question, as I do not feel like I have a sufficient answer.
If I understand Solomonoff induction correctly, given two hypotheses A and B, s.t. A and B both have accurately predicted data so far, the shorter of the two hypotheses will have the higher probability of producing a correct result for the next prediction. In other words, the longer one will have a lower probability of predicting a correct result. This is the same as saying that it is more likely to be falsified.
It seems that part of the problem in answering this question comes down to a disagreement between what constitutes a unique theory. I would call y = 2x and y = 3x distinct theories. Yes; they fall into the same class, but they are not the same theory. The same is true with y ∝ x. y = 2x and y = 3x are similar theories: it's that theory with an additional assumption: the specific constant proportionality is either 2 or 3.
There are some good reasons why simply varying the parameter should be considered a different theory. For one thing, suppose we have a general power series as our theory. By varying different parameters, we can get any single function which is complex differentiable on an open set. That would include y = e^x, y = 8sin(x), y = 3x^2+2x+1, etc. But clearly those are modeling fairly different phenomena.