Is the lawyer's fee for filing for probate an acceptable legal fee?

[tagteam]

It is if the fee is paid for by the estate.

If you paid the fee out of personal funds, the estate should reimburse you.

[Opus 17]

First, any tax deduction can only be taken against taxable income.  You can’t deduct personal legal fees that are unrelated to taxable income.  

If the estate has ongoing income (like a writer or musician), legal fees to administer the estate might be deductible business expenses against the ongoing income.  However, most personal estates (like your parent) will not meet this test, and legal fees are not deductible on your personal tax return and will generally not be deductible on the estate tax return if the estate is required to file a return.  

As said, the estate should pay any legitimate expenses incurred by the administrator before dividing the remaining balance among the heirs.  

[tagteam]

Costs (including legal fees) that that are paid or incurred in connection with the administration of an estate that would not have been incurred if the property were not held in such estate are fully deductible on Form 1041 (or an estate tax return).

See https://www.govinfo.gov/content/pkg/FR-2020-10-19/pdf/2020-21162.pdf