This is 2520 miles - a 38 hour drive without traffic. So it isn't possible to limit the days to 9 hours. The fastest route would have you take I-20 and I-22 from Atlanta
up to Memphis
and then take I-40 all the way across to Bakersfield
and then I-5 up to San Francisco
When I attempt to break this down into four segments of equal length, stopping at places which might have pet friendly motels, the best I can do is Atlanta to Fort Smith
(10 hours) and then to Santa Rosa (New Mexico)
(9 hours), to Kingman
(8.5 hours), to San Francisco (9.5 hours)
I don't know what you mean by 'good parks mid drive'.
If you are driving 9 hours a day, you will have no time to stop at any parks, and you will have to figure out the pet friendly hotels on your own. Use booking.com or google maps.