Your parents can claim the credit since they own the home ... to reduce the cost basis of home you must own the home.
per the IRS:
Who Can Take the Credits
You may be able to take the credits if you made energy saving
improvements to your home located in the United States in
2021.
Home. A home is where you lived in 2021 and can include a
house, houseboat, mobile home, cooperative apartment,
condominium, and a manufactured home that conforms to
Federal Manufactured Home Construction and Safety
Standards.
You must reduce the basis of your home by the amount of
any credit allowed.
