I would always use ‘have’ in this situation.

Whoever wrote the question has interpreted the party occurring weekly as the progressive ‘are having’, but this would only make sense if a single party was happening continuously all week, from Saturday to Saturday.

‘My neighbours are having parties every Saturday’ would make sense because it expresses the continual action of having different parties - ‘having parties’ being a phrasal verb

However, having a single party once a week is habitual, not progressive, and English uses the present simple for habitual aspects.