Test if a string is parenthesly balanced
Japt v2, 20 bytes

V="()"V¥iU1 eViV²1 V


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Everyone misunderstood the challenge at first and though that each pair of parentheses had to contain an even number of sub-pairs, when in fact the challenge actually asks for 0 or 2 sub-pairs. So here's my revised answer, using the same technique as before.

We can still solve the challenge with recursive replacement. The thing is, instead of just removing all occurrences of ()(), we need to make sure that there's nothing else in the same wrapper besides the ()() (in other words, no ()()()() or anything like that). We can do this by recursively replacing (()()) with ().

The new problem is that the input itself does not have one pair of outer parentheses (as that would make it not a parenthesly balanced string), forcing us to wrap it in an extra pair to fully reduce it. Finally, the end result for balanced strings is now () instead of the empty string, so we check for equality rather than just taking logical NOT of the output.