There are no par 2 holes because every hole’s par includes two putts on the green.
I know of several par 6 holes. I play one relatively regularly at The Farmstead Golf Links in Calabash, North Carolina. The 18th hole is 767 yards (tips) par 6. The tee box is in South Carolina, and the green is in North Carolina. Sort of gimmicky, but the hole is enjoyable and actually not that difficult (it’s the number 8 handicap hole on the course).
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That par-6 on No. 18 is a killer for an 18-cap, since it can take their score down significantly. A shorter tee shot might put them short of the fairway, making their second shot more difficult. Water hazards intended to be cleared on a single stroke must be “laid up” in front of.
I play the white tees, so the hole is 679 yards long from those tees. I shot par on the hole the last time I played it.
Holes with long gaps in between them make a course’s “slope rating,” which is a measure of how much more difficult a high-handicap player than a scratch golfer. While certain slopes are to be expected (you can’t really make a course easier for a high-cap than a scratch golfer), extremely steep slope.
Par-6 holes, on the other hand, are more of a spectacle than anything else. They’re found at luxury courses that people go to on vacation just to play golf, making them a “bucket list” item rather than a hole you’d want to play every weekend on your own course.