I highly recommend exploring the following official sites for the two parks for campground information:
YellowstoneGrand TetonFor hiking we really enjoy the Tetons better. You should really plan on staying in both the parks as they're both completely different.
We stay in the parks' no reservation campgrounds - Gros Ventre or Colter Bay for the Tetons (each has 300 non-reservable sites with water fill and dump stations but dry camping) and we like Mammoth for Yellowstone. There's also Fishing Bridge if you want hookups in Yellowstone or Colter Bay RV park (right next door to the non-hookup, non-reservation Colter Bay campground) if you want hookups in the Tetons. Madison is nice for reservations but no hookups. Then we don't have to be on schedule. There are plenty of other campgrounds in the parks that accept reservations.
If coming from the south through the Tetons first, stop just before entering the town of Jackson for major grocery stores - Smith's on the left and Albertson's on the right. That's your last change for decent-size groceries except for tiny, overprices camp stores or in West Yellowstone which has 2 mom & pop small stores.
Your parking within the parks will indeed be tight but it's doable - we've seen others doing it. It wouldn't be our way of going though. We tow a car. Even renting a dolly for your Taurus might be worthwhile.
For Glacier, stay at Apgar c.g. in the park and you can easily walk to the free shuttle bus for your trip up Going to the Sun Hwy with many hiking opportunities along the way. Just get off wherever you want and hop back on the next bus.
Have a great trip. It's a hard one to do with limited vacation time but you just do what you can. We've all been in the same boat.
If you main goal is visiting the 3 national parks, don't waste time beforehand stopping at other places to tour. Finish those 3 parks and on your return home, if you still have time, then you can check out other places. I would bet you won't have time. Those 3 are large and time-consuming!
Full-Timed for 16 Years
.... Back in S&B Again
Traveled 8 yr in a 40' 2004 Newmar Dutch Star Motorhome
& 8 yr in a 33' Travel Supreme 5th Wheel