I recommend you visit the NPS websites for each park on your list. Pull up the "things to see and do" list for each park. Sit down with the family and go through the lists checking off each item any of you wants to see/do. Once you've done that for each park, break those items into two groups:
1. Those that can be done in 3 hours or less, and
2. Those that will take at least 3 hours
Now, plan on two items from group 1 per day and one item per group 2 per day. So, if you have 14 items from group 1 and 10 items from group 2, you will need a minimum of 17 days with no driving to see what you KNOW you want to see. That will allow for some downtime, meals, etc.
What I would recommend is to pick two of the smaller parks and plan to stay at each for a week with reservations. Then set some unreserved days before the first park, between the two parks, after the second park, and in Seattle to explore the areas and what's outside of the parks.
Save Yellowstone for it's own trip and spend time to enjoy it as much as possible. You have an RV, it's not like this is the only trip you can take. Slow down and see each park as fully as possible rather than rushing throught them to say "we've been there" and be too tired and stressed to enjoy it.
Think of it this way, your trip plan is like "It's Thursday, we must be in Brussells". It will be like a bus tour trip. Get on the bus, drive for hours, get off quick, see the Mona Lisa, hurry get back on the bus, drive for hours, get off quick, visit the Vatican, oh shoot we're out of time we didn't even get to see the inside, hurry get back on the bus...
Can it be done - sure. Do you REALLY enjoy your trip? Do you regret not being able to spend more time enjoying Paris and Rome?
1975 American Clipper RV with Dodge 360 (photo in profile)
1998 American Clipper Fold n Roll Folding Trailer
Both born in Morgan Hill, CA to Irv Perch (Daddy of the Aristocrat trailers)