I would do:
Black canyon 2 days
Mesa verde 2 days
Great sand dunes 1 day

What time of the year are you going? Keep in mind all three of these parks can get extremely hot in the summertime and you’ll only be able to do hikes in early morning or after sundown. 

In terms of lodging, I can recommend some great camping but if you’re not much of a camper then I can’t help you there

I just did these three parks this past week! A day for each park was perfect for me.

Sand Dunes: I rented sand sleds and spent half a day sledding. You could make it a full day by also hiking to some of the higher dunes.

Mesa Verde: I did one dwelling tour (Long House). It was an awesome experience. I’d recommend a second tour as well to make it a full day.

Black Canyon: Spent one day on the south rim drive, going to the overlooks and the short hikes associated with them. If you had two days you could also go to the north rim.

Hope that helps!

I would plan on spending some time in Ouray on your way through.  A few good waterfall hikes, Box Canyon is excellent.  If you're adventurous the Ouray via ferrata is a lot of fun.

Did you go on this trip yet? How was it in 5 days of travel? Do you have any tips as we are attempting the same trip in early June

I’ll have five days to visit these three parks. I’m curious about your thoughts on how much time to allocate to each park and/or if there are other things in the area I should also check out. Any recommendations on lodging options near these parks? Thanks for any suggestions you can share!

(Question) How much time would you allocate to Black Canyon of the Gunnison, Mesa Verde, and Great Sand Dunes national parks?