We've had several inches of rain last night and this morning.
I've noticed my septic sprinklers have run a few times already today, however we are not using any water. I've verified none of my toilets are running or any faucets leaking. I opened the third section and could see water draining into that section.

I guess my question is could the rain just be leaking into my septic tank from all the rain and that's what's causing the sprinklers to come on or do I need to be more worried about a leak somewhere?

More than likely your tanks have settled and are too low if rainwater is leaking causing the heads to spray. You may need to have risers added to the tanks and or adjust current grade/drainage.

The concrete top is a couple feet below the ground. But the round entrance lids are still above ground.

Sounds like maybe where that connection is, is leaking? Is that a problem other than my sprinklers running more often than needed.

Running the pump more often will shorten its life and waste electricity. Also if the rainwater is taking debris into the pump tank it can burn the pump up sooner.