"The party yesterday was fun."
The same thing. As far i know there is no way to construct the sentence in such a way that it is analogous to "the party yesterday" in English.
How about "yesterday, the party was fun"? You don't necessarily need to bestow ownership on yesterday. It means the same.
In casual speech is fine if a short pause is taken, but would sound weird, in casual writing, a comma would suffice
Is the "の" particle really needed here? Surely 昨日、パーティは楽しかったです。would mean the same thing?
Because the i-adjective 楽しかった（たのしかった）is what indicates past tense here. The ~かった ending means "it was ~". Since this is the basic/informal form, です is added for politeness and does not change tense itself. This kind of makes it sound like "[the fact] is, it was fun" (both a present and past tense in one sentence).
でした only comes into play when the i-adjective cannot make a paste tense of itself, as happens with formal negatives. E.g. when instead of saying 楽しくないです you use 楽しくはありません, the latter can't be conjugated any further. So an average polite 楽しくなかったです becomes a highly formal 楽しくはありませんでした.
Shouldn't 「きのう、パーティーがたのしかったです」be fine?? Id think that if they really really wanted me to say 「きのうのパーティー」then they would have asked me to type "yesterday's party" in the question...
All of duolingo has taught the polite form so far. If you remove the desu it would be technically correct, but not the polite form.