ok i did a brief survey and this is what i got so far: there are two words for wedding: "mariage" and "noces";
Il l'a demandée en mariage. (He asked her to marry him.)
mariage religieux (church wedding)
mariage de couleurs (blend of colors)
noces de diamant (diamond wedding)
repas de noces (wedding breakfast)
voyage de noces (honeymoon trip)
The French is "de mes parents" (plural), so it will be « parents' ». Unfortunately, a gremlin is loose in the system who has been allowed the authority to insert incorrect apostrophes without the knowledge to do so correctly. Until the little dude is removed, we all suffer the consequences.
See the other comments. There's an error in the programming. The system hides the apostrophe by mistake. It's not the moderators' fault. Otherwise, no, there's no difference between AmE and BrE when it comes to possessives. The correct form is "My parents' wedding", i.e. "The wedding of my parents".
Typically, FR "parent" will mean "relative" or "relation", but when used with a possessive adjective, e.g., mes parents, it will be understood as "my parents". It's the same with "la femme" (the woman) and "ma femme" (my wife); "la fille" (the girl), "ma fille" (my daughter).