Pinch flats are due to under-inflation of tires. On a road bike you should be inflating your tires to 120-130 lbs.
Also, you need to check and re-inflate your tires every time you ride. I just keep my pump right next to my bike, and wind up adding air every 2nd or 3rd day.
I see you added tire size - those are older bike tires, they don't inflate as high as road bike tires, I'd try pumping them to 65 lbs, and if you are still getting pinch flats, go up 5 lbs at a time until it stops.
You can tell a pinch flat because it will look like a snake bit, 2 small holes close together.
If you do not get any pinch flats at 65 lbs, you can try going down 5 lbs at a time until they return, then you know the range your tires need to be at to be properly inflated.
The most important thing is to check tire pressure every time you ride, as bike tires can lose up to 5/lbs a week from just sitting. These are high pressure tires, so they lose air much faster than low pressure car tires.