Your dad can file his tax return using the Head of Household filing status if you file your taxes separately from your spouse. pay more than half of the household expenses, and not have lived with your spouse for the last 6 months of the year.
When you say they cannot amend to filing Married Filing Jointly because the tax will change, there really are only two options available to them.
First, mom can file Married Filing separately consistent with dad's filing Head of Household even if that results in them paying higher tax between them both.
Or Two, amending dad's return to be Married Filing Joint again even if it means paying higher tax if the Head of Household filing status was not correct in the first place.
Not amending the tax return because the result is a higher tax is not a valid reason for not amending a return to correct an error on an tax return.