Hi,
If your mum is a beneficial owner of the property and will recieve a share of the disposal proceeds, then she may have a capital gains liability from the disposal.
If your father will retain all of the disposal proceeds as the beneifical owner, then he will have to work out if his Private Residence Relief covers all of the gain. You can see guidance here:
HS283 Private Residence Relief (2024)
There is a calculator for capital gains, which links into reporting and paying any Capital Gains Tax due, here:
Tax when you sell property
Thank you.