How to earn Party characters?!?!

What you'll first need to do is complete the solo quests for either the Plant or Zombie side; the "L.E.A.F." or "Top Zombie" quests. Completing either one unlocks Infinity Time for that side.

Once you've unlocked Infinity Time you'll need to play through it, and play through it well. Your best chance to receive a piece of a Party Time character variant is to collect at least 25,000 Time Shards which qualifies you for a prize from the Infinite Infinity Chest- which essentially means getting up to or past Wave 25 either by yourself or with a team of up to 4 Players including yourself. Getting vanquished with fewer Shards can still get you a character piece but the chances of that are reduced.

When specifically questing for the Party variants you should consider letting yourself get vanquished after you've passed 25,000 Time Shards because getting more than that will not increase your chances of getting a character piece or the number of awards you will get so there's little point in continuing unless you're trying to see how far you can get.

Pieces of the Party Time characters can also drop from the Infinity Time Sticker Packs which appear once in a while in the Sticker Shop for a limited time and when Rux appears in his shop in the sewers once a month he could possibly have a character piece to offer for a large price tag since he specifically stocks things that you don't already have.

The "fastest" way to get the characters unlocked is to play a lot of Infinity Time and to not get discouraged by the fact that you probably will need to invest a lot of time and energy into it before it pays off with what you're looking for instead of some customization item instead.