I don't think that this is what Jen needs to worry about.
If they really wanted to have her belted 24/7, they simply could have refused to open the belt one morning and told her "get used to it".
And put yourself into the position of her parents. They have probably other priorities in their life than thinking about what after exam party Jen has planned. Now that party is getting closer and with the old car becoming available now, the question what to do with it came up. It might be on short notice for Jen, but from her partents perspective it was most likely just what came up naturally at this time.
And I don't think it is unfair from her parents to make that offer.
If Jen rejects the car, nothing in her life has changed. She can party just the way she has planned. So why is that unfair then?
Of course parents are worried that in such a situation self-control over the key will not be good enough for the same reasons that I have doubted that in general this self-control will work until marriage. Both her parents and I might be wrong, but I bet they have their own life experience and based on that they want her belted more strictly at least during such high-risk events and then they just came up with the idea of offering her the car in return for accepting that additional control.