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Hello, I will assist.
As you are at home and paying rent, you are classed as a Lodger.
Your parents are within the law to ask you to leave, if you fail to, they can evict you without a court order.
The amount of notice they must give would depend of the rent you pay. If you pay weekly, then they must give you a weeks notice in writing. If you pay monthly, then they must give you a months notice.
If you are named on the tenancy agreement then no.