Solitary hiker,
I did part of this route, but from the other direction a few days ago. Some of the trail between Mt. Parker and Mt. Resolution was a bit overgrown but I think adding Mt. Resolution would still be a manageable day hike; I guess that's all perspective. I put a description of this in the trip report section.
I think that it is just under two and a half miles from the Parker summit to the Resolution summit, either one would be a good lunch stop with fine views. The 3 miles that Pilgrim touched on will take you from the Parker summit to the Resolution shelter which is a bit past the Resolution summit from the direction that you would be approaching from. The Langdon Trail is pretty flat and easy going.
Enjoy and happy hiking.