I'd probably rank them Karlsson >> Parayko > Burns > Pietrangelo. I think Parayko, Burns and Petro are all in the same tier though and rankings can be a bit subjective within tiers since there is some degree of uncertainty. It's a weird comparison since Burns and Parayko are such drastically different players. Burns 17/18 season was rough and if you feel it as a total outlier I think it's reasonable to say he is uber-elite offensively. I wish I had access to Micah's data (maybe I'll cave in and pay for it at some point), and I don't pay a lot of attention to micro-stats so don't really have anything to say about that. I do see a large gap in are point rates and corsi, but neither of those stats sway my opinion all that much tbh. I guess an advantage could be that you can play Burns with forwards who are better at driving play but worse scorers and trust that he can drive scoring with goals and shot assists. With Parayko I agree that the metrics pretty much accurate, though I think he can provide an above-average offensive impact when you don't ask him to carry the offensive load if that makes sense. When he is with Bouwmeester and asked to be the defenseman that is more active in the offense he isn't great because that doesn't fit his skillset, but if you give him a guy like Dunn and let him run the show while Parayko supports and uses his great shot I think he adds some value. I also think you underrate Petro a bit when looking at just his single season statistics. Over the last 3 years his RAPM and GAR are very impressive (on EvolvingHockey's site at least I hadn't really used corsica's GAR up to this point so I'll have to look into that).
I would strongly suggest caving and paying for access to Micah's stuff. I personally think that EvolvingHockey's RAPM is superior to his model, and especially so when judging defensemen, but it helps to have another opinion and it's also very good for dispelling narratives and explaining things to people who aren't too into stats. For example, look at this:
(Bonus: Sometimes heat maps look like the mask from Scream.)
This is a pretty strong retort to the "Binnington carried the Blues" narrative, which I'm sure you are just as sick of hearing as I am. And I think it helps get the point across better than just posting xGF/xGA or Corsi or whatever, since people tend to be more receptive to images than numbers. Even me being a super stats nerd, seeing WOWY heat maps for shot against and Connor McDavid helped challenge a pre-conceived notion I had that McDavid hadn't done significant damage to his team on the defensive side of things.
Anyhow, back on topic, I think your assessment is pretty fair. I might be underrating Pietrangelo a bit as he had one unspectacular regular season but followed it up with an excellent playoffs, and he has a track record of being excellent. I would still take Burns over Parayko just because I've seen how Burns can carry my team into the playoffs as the offensive leader, and that's extremely valuable. I do think that the 2017-2018 season was an aberration for Burns where poor shooting luck early in the season kind of tanked his confidence and all-around game, and if you look at his last 4 seasons, you've got 3 elite ones and then just one where he wasn't good. We'll see what happens next year.
And yeah it feels weird to say Parayko is just average offensively because he has that wicked slapshot, and he is a little above average offensively in a 3-year sample so maybe I'm selling him short. I do think he probably deserved to go higher on this list.
Also Corsica's GAR is pretty bleh. Joe Pavelski was a top-5 skater this year and Logan Couture was in 2017-2018, but I can promise you neither were even remotely close to that good in those seasons. It puts too much emphasis on shooting performance IMO. But it's just another piece of evidence that I try to take into account when looking at players just because it does exist and is objective.