Once you are deemed to be a resident, you're not allowed to drive a car registered abroad (except if the owner of the car is also present), otherwise it would be too easy to circumvent all the other obligations. Consequently, you will not be able to drive with your parents' car while residing in France (see also europa.eu).
If you would be the owner of the car, you have one month to register it in France after moving. And if you buy a car abroad, the rule is the same. So transferring ownership before importing the car will only change the list of documents you have to produce to register it, not how long you will be allowed to drive it. That much is easy.
Now, you need a bunch of documents to register the car in France. The full list is on service-public.fr, but only in French. You will need some proof of address, ID, registration certificate for the vehicle, a couple of forms (your parents can sign one to confirm they want to give you the car), a European type-approval certificate (if you don't have one, you get it from the manufacturer of the car, possibly for a fee) and a roadworthiness test results (you have to pay for that one too, obviously). You also need a “quitus fiscal”, which you can obtain from the tax office.
How much the registration itself costs depends on many things and in particular on the exact vehicle but also where you live (there is a regional tax) and the composition of your family (there are discounts for people with many children).
I am not aware of any comprehensive explanation in English but you are expected to provide everything in French in any case (the authorities can actually demand a written translation) so managing the whole process without enlisting the help of a French speaker will be very difficult.
