。
She insisted I go to the party.
Literal
She [topic-は] I [subject-が] that party [to-に] go [as-ように] insisted.
主張する ('insist, claim') with the ~ように-marked indirect command is a standard pairing for forcefully reported speech: she didn't merely suggest the party — she insisted. パーティー is a katakana loanword from English 'party'; native equivalents like 集まり ('gathering') feel quainter and less Western. その ('that') here is anaphoric — pointing back to a party already established in context, not physically pointed at. が marks the embedded subject 私.