I don't know what else to call it. It ought to rank up there with the latest addition to the dictionaries along with 'humblebrag' and 'backhanded compliment', among others.
It goes like this.
Proud parent "My child just got accepted into an exclusive program at X university for pre-med!"
Listener " Oh, wow, congratulations! You must be so proud of her. That's a great school. I know because my aunt's second cousin's niece graduated from there as a neurosurgeon with a 4.0 GPA at the tender age of 20!"
Proud parent looks deflated, Listener is blissful at having been able to suck out all the air from the former's sense of pride in her offspring.
For the want of a better description, I propose 'selfie compliment' as a name for this mode. The object is not to shower compliments on the 'complimentee'; rather, its purpose is to drag them down from their lofty pedestal while reflecting the utmost credit on the family pride of the 'complimenter'.
Highly common in competitive societies (ex. Indian, Jewish, Chinese ), this would sit well in academic descriptions like the one I found in Google books about The Primitive Mind and Man by John Alan Cohan about the giving out of compliments and the reactions to them.
Why can't a compliment be a simple appreciation of something that someone has done, rather than a mode to lessen their happiness by triggering envy?