Their payments would stop because they would repay the mortgage with the money you paid for the house.

lots of implications....

1. You cannot become sole owner unless parent's mortgage is paid off in full. How much is it seeing as it is interest only so has never reduced? (What they can buy the property for is irrelevant in terms of clearing the mortgage)

2. if you become sole owner and they continue living there they must pay you rent otherwise they will be caught by the pre owned asset tax rules which result in them being exposed to paying income tax on the notional value of the rent the property would get on the open market. The same applies if they pay you a discounted rent, they either pay full market rate or they get taxed seeing as they have sold their asset to you at a discount.

3. Continued occupation of the property by its previous owners will be a stumbling block for many lenders as the parents could be seen to still have some "rights" if a repossession was required following you defaulting on your own mortgage - ie. they may not want to lend to you at all

4. You would be a landlord with all the implications that brings. However, as it appears you intend to reside there yourself your parents would be your lodgers rather than your tenants (unless you do something stupid like given the exclusive use of a part of the property and this create a tenancy).

5. Of course you will have to pay tax on the rental profit you make - it is money you receive which has not been taxed ! However, with lodgers you can then claim the rent a room allowance which may more than cover the rent they pay you (£7,500?)

6. Whether a lender would include that rental income as part of your overall affordability calculation is a very moot point seeing as the source is your parents, ie: the ex owners and ongoing residents with zero chance you as their LL would evict them for non payment of rent and replace then with strangers who would pay rent. Many lenders will shy away and base your affordability excluding the rent.