Squeaking when rocking car back and forth

So, replace the struts. Do you find that this is implausible? How many miles on the odometer?

 

I am wondering about your OP. If the car is parked, why would you push it by hand?

In any event, at 112K miles it doesn't surprise me that the front struts and rear shocks would need to be replaced.

However the cost of doing that is high enough so that you'll probably need other reasons to take action before you'll decide to make the investment. For example, if you find that when you go over bumps, the car continues to bounce after you hit the bump, that is a good indication that the struts/shocks are worn. If you see that the tires are not wearing evenly, that is another good indication that you have worn suspension parts (or maybe the car needs a wheel alignment.)

If you drive a new Prius and find that the ride is so much better than your car, that is a further clue that your car might benefit from new struts/shocks - although you can't expect the ride to be fully restored to like-new.

 

If that is the only issue, I think you can safely ignore it. Press hard on the parking brake pedal when you park the car, and see if that helps.

 

Did you ever figure out what the cause of the sound was? I have seen some simlar videos saying it was the exhaust downpipe gasket. Just curious if you did anything about it.