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A Reddit user registered the domain name slutsofinstagram.com, and has created a kind of parody cover story (Reddit; Imgur):

Welcome to Slütsof In Stagrâm an online fantasy series about a Princess duck named Slütsof and her adventurous journey across the mystical lands of Stâgram.

Instagram allegedly sent this user a letter (image) asking them to which alludes to the Anticybersquatting Consumer Protection Act (ACPA). As a lay person, I cannot determine how the ACPA would apply in this specific case (if at all), and in particular it seems to fail the condition:

... has a bad faith intent to profit from that mark ...
_{15 U.S. Code § 1125 - False designations of origin, false descriptions, and dilution forbidden}

Question: On what grounds can Instagram contest the use of the domain name slutsofinstagram.com?

I found Trademarked name in domain name, but the answer there seems practical: they can just bully to get their way with expensive lawsuits. I'm interested in the theoretical here: if they have actual grounds.

The theoretical is that the domain slutsofinstagram.com does dilute or harm Instagram's mark because of the use of "Instagram" in the domain and parody and because of the unwanted association of the parody story; that's the theory and "grounds" that Instagram would use in their lawsuit.

Instagram would have to prove their theory of dilution or harm to the jury or judge, and convince either that the harm is not theoretical but real and financially damaging in order to win their case. The owners of slutsofinstagram.com would have to defend themselves and argue that they are not damaging Instagram's mark and their parody is a parody and is protected as such.

Any actual court outcome is also theoretical. If Instagram won their case, one of the legal remedies could be the court ordering the owner of slutsofinstagram.com to transfer ownership of the domain to Instagram so they could simply park the domain and effectively take down the parody story. Or, the owner of slutsofinstagram.com could prevail because they argued the fact - and the court agreed - that the domain name and creative contents of the site is a parody and is protected speech and doesn't dilute Instagram's mark because any reasonable person could see that the site is a parody and not related to Instagram.

In the US, a possible example of a parody not diluting a mark is L.L. Bean, Inc. v. Drake Publishers, Inc., 811 F.2d 26 (1st Cir. 1987), where

Imitation may be the highest form of flattery, but plaintiff-appellee
L.L. Bean, Inc., was neither flattered nor amused when High Society
magazine published a prurient parody of Bean's famous catalog.

The reality is that Instagram can contest the use of slutsofinstagram.com because they can; they have deep pockets and can take the owner of slutsofinstagram.com to court, like any other trademark holder can contest another's trademark. The owner of slutsofinstagram.com would have to defend themselves in court, or settle - possibly for monetary damages or the ownership of the domain or use of slutsofinstagram - and not go to court. Basically, one man's litigation bully is another man's trademark protection hero.

As always, your mileage may vary due to jurisdiction and national/international laws and agreements concerning trademarks.