Directly addressing the OP question, yes you can do both the PPL and COMM at the same time, but your COMM grade is likely to suffer on the COM, and is a permanent record.
I took Commercial ground school while training for PPL simply because it was available.
Before I completed the course, my CFI signed me off for the PPL written test, I passed easily, and took my PPL check ride.
I completed the COMM course, passed, and received the paper to take the FAA written test.
A few years later, I signed up for both COMM and Instrument training, took both ground tests the same day, and passed both, then a few days later, took the two check rides on consecutive days, and passed both.
So, obviously, I studied for the COMM written test while studying for the PPL written.
I studied for the COMM written test and Instrument written test at the same time.
I studied for the COMM and Instrument flight tests at the same time.