I thought a pitch fork would help to aerate the soil without disturbing the
roots too much.

A spade or a hoe has the risk of slicing thru the root system.

A pitchfork on the other hand, whilst it may hit some roots, would leave it
intact.

My problem is that my soil has a lot of clay. The bad drought that we have
been experiencing has caused the soil to harden. So I need to poke holes
into the soil to allow for more water to get down there instead of just
washing off.