Yes, you may stay in the UK provided you meet the requirements of the next Visa application. To do this, you must be in the country with the Family Visa to the UK for Parents for a qualifying period (usually a minimum of 5 years) before you receive the permanent right to reside (‘indefinite Leave to Remain’). Before extending your visa, check how long it will take for you to settle in the UK.
If you have lived in the UK for many years, you can apply to remain on the basis of your long residence.