Simon - I'm sorry to read of your losses
When a joint owner dies probate is not required, whether that are TIC or Joint tenants. The legal ownership of the property passes to the surviving owner to deal with.
Probate is though required for a sole owner when they die as the property/legal ownership does then form part of their estate.
From a purely registration perspective you will need your Mother's death certificate and Father's probate in order to sell
The second issue is the TIC and if that is protected by a form A/joint ownership restriction on the register. If it is then it will catch any sale as you are a sole executor acting on behalf of a sole deceased owner.
You have two options with one being as the solicitor advises. Appoint someone else to act with you as a co-trustee and together you complete the sale/transfer
The second option is to apply to cancel the form A restriction using forms RX3/ST5 or your conveyancer can provide a certificate re the relevant facts if appropriate - see PG 6 section 8 for the guidance on the facts needed
Practice guide 6: devolution on the death of a registered proprietor - GOV.UK