Regarding parentheses and the end of sentence period, when the parentheses begin in 1 sentence but end after a second complete sentence?

“Jim likes bananas (as well as apples. He also likes most vegetables).”

The finished sentence in the beginning of the parentheses, with 1 or more complete sentences after the partial sentence, all within the same parentheses. Where does the final period go - in or out of the closing parenthesis? Thank you!

Any sentence with parenthetical information needs to be constructed in such a way that the optional information makes no difference to the main part of the sentence. In other words, if by removing the information the sentence becomes ungrammatical, then the sentence is ungrammatical and the parenthetical information has been misapplied.

For instance:

✔ Jim likes bananas (as well as apples. He also likes most vegetables).
→ Jim likes bananas.

✘ Jim likes bananas (as well as apples. He also likes most vegetables.)
→ Jim likes bananas

This applies to syntax and semantics as well as to punctuation:

✔ Jim (sometimes) likes to eat (ripe) bananas.
→ Jim likes to eat bananas.

✘ Jim (sometimes likes) to (eat ripe) bananas.
→ Jim to bananas.

As for rules governing how information inside parentheses should be presented (partial sentences, mixed sentences, and so on), that's a matter of style.

Many style guides say that if you break the information into multiple sentences, the so-called first sentence should start with a lowercase letter. Some indicate that the final sentence not have a terminating period. In order for the main sentence to be correct, a period has to go outside the final parenthesis. Whether or not this results in using two terminating periods, one inside and the other outside, or just using one—outside—might be a grey area, stylistically speaking.

If it can be presented in a way that appears simpler, or avoids this confusion, it's probably better to do so. (As one comment under the question said, use a semicolon rather than a period in order to keep it all as a single phrase so that it doesn't look odd.) In the case of this particular sentence, I personally don't see a reason to use parentheses in the first place; however, I acknowledge their use may have been contrived simply for the sake of the question.

Regardless, as a matter of style, how the information inside parentheses is presented is subjective—or determined by the particular style guide being followed.