Hi Justyna,
HMRC doesn't count cash gifts as income, so you won't have to pay any income tax on cash gifts received from parents (or grandparents for that matter).
However, if there is interest earned in a savings account, you may be liable to pay tax on it.
Please also farmiliarise yourself with Inheritance tax guidance, this is seperate to Income Tax -
How Inheritance Tax works: thresholds, rules and allowances
Thank you.