Question

why did mrs and mr bramble decide to keep bramble profession a secret from harold?

Answer 1

as they were of thought that Harold will feel embarrassed if he came to know the occupation of his father

Answer 2

Harold was a child prodigy and Mr Bramble belonged to a rough business of boxing. Mr and Mrs Bramble worried that if they would reveal the secret to Harold, it may hurt him or affect his studies. That's why they wanted to keep Mr. Brambles profession a secret from Harold.