We have never once had to park the stroller in any TS restaurant. When we checked in at the counter, we simply told the hostess that we had an infant who could not sit in a high chair yet (DS was 3 months, DD was 9 weeks at first trip and 4 months at second trip). We don't use the infant car seat while we're in Disney and we didn't want to hold a baby while we ate.They were all very accomodating. Here are restaurants we have used the stroller:
Rose and Crown (single Graco stroller)
San Angel Inn (single Graco)
Ohana (double Combi stroller)
Kona (double Combi)
ESPN (double Combi)
Big River Grilll (double Combi) sat outside
Hollywood and Vine (double Combi)
T Rex (double Combi)
Chef Mickey's and Crystal Palace both had infant high chairs which were great! DD will be 6 months old when we go in a few weeks and we are still not sure what we will do. She sits up by herself and will start solid food by then. Our high chair at home has a back but the Disney ones do not. We may take blankets or a big towel and prop her up.
You will be fine, just mention that you have an infant when you make your ADR's.

I also said I would be bringing a stroller into the restaurant when I called, but the lady on the phone each time said just to mention a stroller at check in.