They know who X and Y are, and also know the identity of their son (you), so that gains them your unique DNA sequence, identified as yours specifically.
How do you plan to do it anonymously, considering what you now know?:
1. There are already multiple database containing both your parents, you, and a linkage between you and them indicating parentage. So, prior knowledge: Alice and Bob are parents of Charlie.
2. If Charlie's parents have taken a DNA test, there already exists a database linking their DNA to their name. So, prior knowledge: Alice's DNA belongs to Alice, Bob's DNA belongs to Bob.
3. If Charlie takes a DNA test totally anonymously and perfectly untraceably, it will still show up as, child of Alice and Bob's DNA. So, knowledge now includes: Charlie's (anonymous) DNA is the son of Alice and Bob's DNA
4. From these pieces of information, it is trivial to de-anonymize Charlie's DNA, linking it to Charlie's identity: the only person it could belong to is the son of Alice and Bob, and the son of Alice and Bob is already known from point 1.
I think in my case I'm just not that concerned by the hypothetical because my parents haven't done sequencing/genetic screening and also aren't likely to. I guess the main question is how far out in my family tree I have to think about that. (Also has implications for my descendants, I suppose...)