Day 1: Atlanta – Birmingham – Memphis – Ft. Smith, AR. 9.5 hrs. Day 2: Ft. Smith, AR – OKC – Amarillo, Albuquerque. 10.5 hrs. Not much to see along the way, but you're going to have to traverse the plains one way or another. 9 hrs a day is not realistic anyway to cover this distance. Day 3: Albuquerque – Flagstaff – Barstow, CA. 9.75 hrs. Day 4: Barstow – San Francisco. 6.5 hrs. 4 days does not allow you any time for parks or sightseeing. You will be in the car driving, stopping for gas and/or food, and crashing in a motel room to get enough rest to wake up and do it all again the next day. If you want to enjoy the sights along the way, add in more days. You are traveling through some amazing landscapes but you would need more time to enjoy them. Places I would consider stopping to explore: Beale St., Memphis. Hot Springs, AR. Grand Canyon south rim or Sedona, AZ. Hoover Dam, Valley of Fire, Red Rock Canyon (Las Vegas area). Death Valley and Yosemite or Sequoia/Kings Canyon National Parks.
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.