You're talking to a friend who's feeling depressed and saying negative things about himself. He says that he thinks his parents are disappointed in him. You say this because you don't think that's true at all.
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This means "That's not true!" or "That's a silly thing to say!" The word "nonsense" means something that doesn't make sense or doesn't have any meaning.
One situation where people often say "Nonsense!" is when they're reassuring someone who's speaking negatively about themselves. For example:
A: It's so hard to meet people. I feel like I'll never find a boyfriend.
B: Nonsense! You're so awesome. I bet there's tons of guys who'd love to go out with you.