If your thumbs are sore, you're not doing it right.
Should be using the heels of your palms mostly. I'll try to describe it, but a good mechanic can show you in a moment's time.
As I wrote earlier, start opposite the valve hole, and make sure you are pushing the tire beads into the center channel as you go -- this is crucial, as it gives you more slack. As you get to the last few inches, right around the valve hole, put THAT section of the wheel AWAY from your body, holding the wheel almost perpendicular to your gut. IOW, you're holding the wheel like you're delivering a very large pizza, with the valve hole jammed into your belly button and your arms stretched out in front of you, palms resting on the opposite sides of the unmounted section of tire. THEN use your PALMS to push the last section of tire up and over the rim's edge...You will almost be 'rolling' your palms (really, the heels of your palms, the meaty part, mostly) on the tire bead to massage and push it up and over the rim's sidewall. As each section gets pushed more up and over the rim's edge, your hands are getting closer together until the last bit pops on. Again, you're mostly using the heels of your palms, which means that you get extra traction and protection from your cycling gloves (assuming you are wearing them).
Using your thumbs severely limits your leverage, and makes your success a function of hand strength -- hence
GhostRider62 's comment about not having enough hand strength. But if that is your limiting factor, you're doing it wrong. With the right technique, you're using your arms and upper body to mount the tire.