The fact that someone might make a mistake in doing something does not show it cannot be done. More generally, You seem to be mistaking validating a program's source with the question of whether it performs its intended purpose. These are different things, and attempting to conflate them will only lead to confusion.
Then your mistake appears to be in failing to see that your perceived edge case does not invalidate the first sentence of my reply. If the sole purpose of a parser is to syntactically validate its own source (which is not the case for a compiler's parser, by the way, not even if we expand 'its own source' to 'arbitrary input'), then if it does that correctly, that's all there is to it.
It is a general rule that those who avoid answering a question do not have an answer, and this is no exception. Here, You completely misunderstand Chomsky’s hierarchy: By your inverted-hierarchy argument, the simplest regular language would be complex enough that incompleteness would be an issue in its validation.
