Mt Rainier National Park (my rating B)

Mt. Rainier is simply majestic.

If "dog friendly" wasn't part of the equation, I'd rate Mt. Rainier National Park an A. 

Yes, it is a National Park, and yes, in general, dogs are allowed on paved paths, carparks, picnic grounds and camping grounds in a National Park.

There are three entrances into Mt Rainier National Park.  They are highlighted in yellow on the map below.

Mt Rainier is extremely popular in summer, often a super long queue at the gates.  Visiting the park with your fur baby means you won't be hiking which means you won't be frustrated to see the carparks at trailheads are over-full.  There are number of places along the road that you can safely pull over and snap a photo with your little pooch, or simply have a leisure walk and admire the view with your little one.

The best views are from White River Entrance (the sign to Sunrise) and from Stevens Canyon Entrance (the sign to Paradise), because the views of the mighty Mt Rainier are in the same direction as you drive, and pull over spots are easy to manoeuvre.  

On a typical summer day, I often try my luck to Sunrise first (follow the orange arrow and road highlighted in yellow), if I see the queue is at the "X" (map below), I turn around and go to Paradise by following the blue arrow on the map below.  You'd get an excellent Photo spot at the Smiley Face on the map below.

Tips:

• It costs to enter the National Park. You might be able to pay the fee at the entrance gate.  The fee information is linked here: Fees & Passes - Mount Rainier National Park (U.S. National Park Service) (nps.gov)
• If you plan to visit three or more national parks in a year, "America the Beautiful" Pass offers good value.  
• If you plan to visit Mt Rainier National Park outside June-September window, it'd be safe to check the condition of the roads, it's not uncommon that some sections of the roads are closed due to the snow. I learnt this lesson a hard way.