I had a woman on my Facebook who was awful with that. Drove me bananas. In fact, I resisted getting on Facebook for years because I knew that's what people do. She is the worst - her daughter is the best at everything and is never at fault when there is a problem. In fact, the mom is so proud of her daughter's antibullying stance and that kid is the biggest bully my daughter has ever known.
So after a particularly awful week of her bragging, I couldn't take it any more, so I started bragging like crazy. About 3-5 posts per day talking about how great my daughter is. I never mimicked or mocked the other mom or child, I simply countered with my own brags. I wondered how she would react. She never commented or liked anything - within a week, she unfriended me! She couldn't take the bragging!
Make sure you're not being hypocritical, which it doesn't sound like you are, and then try that. For every brag she gives, brag back, and make sure your daughter hears you be proud - she could probably use the boost. I usually advocate taking the high road, but this was too fun with great results. And by the way, last week that child, after 6 years of being in class with my child, changed schools. Hallelujah! Is it a coincidence my daughter is coming home all smiles and her quiz scores have been all As since she left?
Eta - this is more than being threatened by someone else's success - this is meant to degrade. Literally everything my child accomplished, the other child did two years before and better - sang on stage? Oh, I remember when my daughter did that... Broke a board in karate? Oh, she did that when she was five! Always said as if relating, but always done to bring the attention back to her own child. And the kid used to say rude things to my daughter, too. I'm very comfortable with our achievements, but the bragging shows that the other mom is not. If you're comfortable, you don't need to brag.