I think that you are confusing two separate tax related issues:
(1) Rent relief: tax relief which can be claimed by tenants -
(2) Rent a room relief: tax relief on up to €7,620 p.a. which can be claimed by people renting a room or rooms in their principal private residence -
Or maybe you mean can you claim (1) AND your parents claim (2)? I can't see why not as long as the transactions are all above board and documented and there is no subsequent transfer/redemption of payments (basically no tax evasion).