If you and your parents both put up money to buy a house or anything else, what share each of you owns would be a subject for negotiation and agreement between you. To the best of my knowledge, there is no law that says "if person X pays the down payment and person Y pays the monthly payments, than X owns 40% and Y owns 60%" or any other specific numbers. Parents often give their children money to help with a down payment on a house or a car with the understanding that this is a gift and the child still owns 100% of the item. Other times they are unwilling or unable to just give the money and want some stake in exchange.
In the case of a house or a car, there's a title that identifies the owner, and legally the owner is the person or people named on the title.
I'd suggest that if you want to have split ownership, like if your parents are saying that they'll help with the down payment but they want to get that money back when you sell or some such, that you come up with a written agreement saying who owns what percentage and you both sign it.
If there was a dispute -- if you never had an agreement about what share each owned and now you're selling the house and you're arguing over how much of the money each of you should get, or your parents want you to sell the house so they can get their money back but you don't want to sell, or whatever -- ultimately a court would decide. Presumably the judge would consider how much you had each paid in, but he might also consider who's been paying property taxes, how much work each has done to maintain the place, etc. It's better to have a written and signed agreement, something that everyone involved is satisfied with and where you all know exactly what you're agreeing to, rather than having a nasty surprise when a judge says no, you're not getting what you were assuming you were getting.