Sorry for you're loss. Use the DL number of the person listed as the taxpayer on the 1040. That would be the parent, not you.
Depending on how long it's been since they passed, do not be surprised if your e-file attempt is rejected with an error of "SSN LOCKED". That just means that whoever handle the final arrangements of the deceased did their job. So the IRS has locked their SSN to prevent fraudulent use. If that happens, then you have to print, sign and mail it to the IRS, along with the 1310.
If tax payer does not have a driver license or state-issued ID (or is deceased), you can indicate that on the return to ful?ll the ID requirement.
Don't know for sure, with a deceased person. It's my understanding that no matter what, if a refund is due to the deceased person, then the living person that will receive the refund has to mail the signed form 1310 to the IRS with their original signature on it. I've never dealt with that myself, so I don't know if that's an absolute fact or not.
Now while there is an option to indicate no ID is available for the tax filer, I do know that tends to delay the refund. But you're dealing with a deceased tax payer. So it's going to be delayed anyway.