Can I sleep in a car in the national parks, like Bryce Canyon or Zion?

Tough question especially with those Parks
which tend to be really popular in summer .
All But Yosemite have really limited campgrounds in the Park and because Yosemite is so close to the Bay Area even it’s available CG’s can fill up early. So make sure that you check with NPS for how early you need to secure a cs in Yosemite

If you’re using established campgrounds and are willing to pay the price then I guess you could do what you are suggesting
I was going to suggest that you do what the overnight hikers do and park your car at the trail-heads parking lots.I’ve done that a lot when I got into a Park late .I’d stop by the Back-country office ,get my back-country hiking permit ,grab dinner and then drive out to the trail-head ,get some sleep in my car and hit the trail first thing in the morning
The only problem with doing that is that your back-country permit application has your license plate number on it and if you’re parked at a trail head for more than a couple of days the w/o a back-country permit on file The NPS rangers may ticket or tow you

This is a little known secret But we have something called National Forests which usually abut National Parks and the rules of camping in them are completely different than those of National Parks are .You can often camp NFs for free and your time limits in the camping areas of National Forests is a lot longer than it is in In National Parks

Since your flying in and out of SLC
Let me suggest another routing than the one that you suggested
Pop this into whatever mapping service you’re using > and yes I have personally driven it
>>>
SLC >South Lake Tahoe CA>Lee Vining CA>Yosemite Village CA >Oakhurst CA>Pinehurst CA>Visalia CA[via Kings Canyon /Sequoia NP] Barstow CA>Willams AZ>GCN>Lees Ferry AZ >Colorado City AZ>Hurricane UT >Mount Carmel Junction UT >Tropic UT> Escalante UT >Torrey UT>Hanksville UT >Moab UT >Price UT >SLC

Since you are visiting all of those national parks
you will need an $80.00 National Parks Pass
http://www.nps.gov/fees_passes.htm
And of course info on the Parks in question
http://www.nps.gov/yose
http://www.nps.gov/seki
http://www.nps.gov/deva ==>> While DV is one of the most stunning places in America ,it’s also hotter than Hell in the summer.The only way that i would do it is if I had a full moon over DV >
If you do seki and find yourself in Bakersfield about 5:00PM then it’s
Bakersfield CA> Weldon CA >Olancha CA >Furnace Creek CA drive right down the valley floor on 118 then >Shoshone CA and if Las Vegas is too much of a culture shock after all that nature then go Baker CA> Laughlin NV [mini shock ] Kingman AZ>GCN
http://www.nps.gov/grca
http://www.nps.gov/zion
http://www.nps.gov/brca
http://www.nps.gov/care
http://www.nps.gov/arch
>>
As far as camping out goes $ 300.00 in basic gear should do it
Campmor is a good and even if you don’t buy anthing from them their catolog is a good source of info to see what’s out there in a lot of different price ranges
http://www.campmor.com/outdoor/gear/Home_
Enjoy America

there are lots of places to stay in sprindale right outside zion. really to access any of the trails you have to take a bus. the road in zion only goes through, no good places to park