Is it possible to contract "would" with plural nouns like: things'd be better if... or My parents'd like it...?

Certainly, however, there are words that do not lend themselves well to contractions of "would".  Is it easy for you to say "things'd" and "parents'd"?  It is not for me.  I would avoid contractions with these words.  The easiest words to use with contractions end in vowels:  I'd, you'd, he'd. she'd, we'd, they'd, etc.

You asked a very good question.  I hope you have more of them.