*** Moved from Bringing Family of USC forum to AOS from Work/Student/Tourist Visas forum -- topic is about AOS of parent already in the US ***
In that case, you and your father will need to submit an I-485 package, along with the I-130 petition. I-485 is the application for green card (GC) while staying in the US.
In addition to the I-130 and I-485, an I-864 affidavit of support and financial documents from you will be required with the initial I-485 package.
I-765 and I-131 are both optional, but recommended with the I-485. I-765 will get your father an EAD card, which will enable him to get SSN, possibly sooner than getting a GC. I-131 is for AP (advanced parole), which will enable him to re-enter the US without abandoning his I-485 process, if he ever needs to leave before getting a GC. Note that there's no fee for I-765 and I-131, if submitted with a fully-paid I-485.
Based on the info you've shared, I don't see any circumstance that could qualify your father for an expedite. So I strongly suggest filing I-131 as a backup in case your father does need to leave the US temporarily. Or scrap the plan to stay in the US while applying for a GC, and pursue an IR5 visa from Canada instead.