I've driven between Seattle and Spokane / Missoula *countless* times throughout my life. I would say that 13 hours to get to Yellowstone-- itself an enormous place-- is pretty optimistic. WA and MT are huuuuuge states. It's 300 miles from Seattle to the Idaho border alone.
With two people you could get there in one good, long-ass day, but you'll have to leave early in the morning and you'd be getting in late at night, even when you're hauling ass. It would be much better to split it up. There are a lot of long, straight stretches in Montana where there is NOTHING but trees for miles and miles, and driving in that when you're tired can be extremely dangerous. People have been known to get what can be called "road hypnosis", and crash into the back of abandoned cars parked on the side of the highway, thinking they're vehicles traveling in front of them.
So make sure you're well-rested!
All that aside, it's a beautiful drive though. Except the part between the Columbia Gorge and Spokane.
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