If the sentence is "our stag and hen parties are on... Date...". Would it be parties, party's or parties'? The parties belong to the stag and hen so I'm presuming there must be an apostrophe? Can anyone help?

I think it would be parties

Parties. You are talking about the multiple of party. There is no possession

Since you already have a grammatically correct sentence with a plural verb (are). The subject and verb must agree---so use the plural form of party.

Parties.