Sounds quite like Tass Times in Tonetown.
Although, IIRC it was about saving Grampa and the antagonist was a kind of a croc with several other besats mixed in 'man', but otherwise it fits. And yes, it was 'punky' :)
It also had a remarkable user interface (that's why I remember it at all) combining the text based nature of classic adventures with a Lucasarts like point and klick structure - except, it had huge icons to klick instead of text. And a rather unique 'copy protection' scheme. To talk to people you had to call them by name - except the names of several important characters where not mentioned within the game (*1), but only in a (real paper) back story booklet packed with the game.
It was available on the II as well as the IIgs, Amiga, Atari ST and many others. On the Apple II graphics where a bit limited, but got as colorful as it can get on a basic II. Way better ofc, when played on a IIgs.
Hardcore Gaming 101 got an article with screenshots of several versions (scroll to the end), which make it look as if there where two sets of graphcs, one for the colourful 16 bit machines (IIgs, Amiga, ST) and one for the less fortunate 8 bit ones (Apple II, C64, IBM PC).
*1 - Ok, many were mentioned during dialogs about them, but never close to where you met them, so it was neccessary to look them up in the booklet.
