Intinerary for a 20 day trip to southern Utah's national and state parks.
 

How many days needed to be allocated for driving,a few hikes and seeing the best of each park?

Which parks are you looking at? I count 10 (Arches, Canyonlands, Glen Canyon, Natural Bridges, Hovenweep, Rainbow Bridge, Capital Reef, Bryce, Zion and Cedar Breaks). Those are just the national parks, not sure how many state parks there are.

Also, any idea on which airport you'd be using or is that dependent on the itinerary you come up with?

When will you be traveling? Several of these will be quite hot in the summer, but you could still hike early in the day or late in the afternoon. Winter can be cold and snowy, limiting hiking in many of the parks. Do you want to spend any time in the back country? Are you planning on staying in campgrounds or hotel/motel/lodge/bbs? How much time will you be willing to drive in a day?

That said, of the ones WhereAreWe mentioned, IMO, Hovenweep, Cedar Breaks, Natural Bridges and Rainbow Bridge could probably be explored in a day or so.

Canyonlands is huge, but most people only visit a very small part of it. The whole Staircase Escalante area could be explored for a few days also.

I think most of what was mentioned above aren't national parks. However, that doen't make them unworthy of some time.

As you can see by the responses, you've got to provide more information.

You could spend a lot of time driving. What's your style of travel. Do you want to relax and see a few parks or do you want to be on the move and see as much as possible? With 20 days you should be able to see most of interest. When?

Im going to assume you mean the entire Southern Utah Region.
To see THE BEST of each park you are going to need to hike in most of them, IMOP.
Zion 2-3 days
Bryce 1-2 days
Capitol Reef 1 day
Goblin Valley 1/2 day
Dead Horse Point 1/2 Day
Canyonlands there are 3 (actually 4)districts 1 day for each
Moab/Arches 2-3 days

Other options
The Wave 1 Day
Lake Powell 1-5 days
Cedar Breaks 1/2-1 day
Calf Creek Falls 3/4 day
Natural Bridges 1/2-1 day

Those are really just the tip of the iceberg. Not sure if you are camping, like to ride horses, hike, 4 wheel, atv or what. Depending on what you like to do will depend on the responses you get. Are you traveling with kids? etc.

You could easily spend a week in each Park. There is that much to see and do in each one. With 3 weeks you should be able to do a lot and I think you will really enjoy it. Do plan it out and have a fairly good idea of what you want to do and how long things take. You don't have to plan it down to the minute, just have a general guideline and then you can make adjustments as you go.

Also Coral Sands State Park (Sand dunes) and Kodachrome Basin State Park

As mentioned before, Coral Pink Sand Dunes State Park is definitely worth a visit. Doesn't take much time, but to see the pink sand dunes. You almost feel like you are looking at the Saharan Desert. We spent less than an hour and didn't feel bad. Kodachrome is beautiful. We wished we would have camped there.

You will definitely and I mean definitely want to drive along Highway 12 through Escalante Nat'l Monument. I could drive it everyday. Breath taking and I am not a fan of red rocks.

Zion is a great place to hike. Definitely want to bring aqua shoes to walk the river up through the narrows. We didn't know and feel cheated on the whole experience. If you are really adventuresome and have a SUV drive the North part of the park through the Kolob Canyons. You will see snow in the upper mountains and see rutted dirt roads, beautiful Aspens--a whole different perspective and opinion of Utah. You will pop out in Cedar City