Would you tell your parents your cat broke a vase if they don't even notice it's missing?

Too bad they found that chip. No I would not have told them. And since they never noticed until that little piece showed up supports you were right. Your father seems paranoid - it is a very odd accusation to make. Wanting to protect a beloved pet is perfectly natural. Broken was an object - lifeless, probably cheaply replaceable and unless this was another in a long series of breakage, your parents should have laughed it off.