You're asking if the inference from There is one A to There are some A's is a logical fallacy. The answer is no, but there is a reason why some seems to mean more than one.
First, note that in your Example 1 the parent can seem to contradict the child even if the parent agreed that there was more than one time:
Child: Sometimes you forget to pick me up from school. Therefore, I want to start walking home.
Parent: I have only forgotten to pick you up twice from school, and that was 3 years ago.
But if the parent can disagree with the child by saying this, does that mean that sometimes means more than twice? No. That's because of the pragmatic, rather than semantic, meaning of the child's assertion. What the child communicates in this context is that the parent forgets to pick the child up from school often enough to warrant walking home instead. That isn't part of the semantic meaning of sometimes, but that's what the child's usage conveys in this context.
The meaning that the speaker conveys that goes beyond the semantic meaning of the words they're using is an implicature. Implicatures are very common in everyday speech. Here is another example using the word some:
A: I ate some of the cookies.
B: (looks at the empty plate) You ate all of them!
Here A's statement conveys the implicature that A didn't eat all of the cookies, although that isn't explicitly stated. Because false implicatures are misleading, B objects to it, even though, again, it's not part of what A said.
How can we tell that the above implicature isn't part of the explicit meaning of A's statement? The answer is that implicatures can be cancelled, for instance:
C: Did A eat some of the cookies?
B: Yes, A definitely ate some of them. In fact, he ate all of them!
Here B's fist part of the statement conveys the implicature that A didn't eat all of the cookies. But the second part cancels it. If not all were part of the semantic meaning of some, B's answer would be a contradiction. But it isn't.
The same goes for some and one. The word some typically conveys the implicature more than one, but not always, and when it does, it can be cancelled:
A: Did you try some of the cookies?
B: Yes, I tried one.
Here B cancels the implicature of more than one. If some logically implied more than one, B's answer would be tantamount to "Yes, no".
One reason why implicatures are generated is that speakers are expected to say things that are relevant and reasonably informative. If you know, for example, that there's only one cookie but say that there are some cookies, you're not being as informative as you can. A hearer would take your statement to convey that there's more than one, not because of logical entailment, but because of the expectation to abide by the norms of conversation. The pragmatic inference goes something like this: if there was just one cookie, you would have said so, but you said some, so there must be more than just one.
