Agreed.
You have to capitalise 'Mom' and 'Dad' here, because they are proper nouns, the same as people's' names are proper nouns. Using lower case would only be possible if you were using them as common nouns, as in 'my dad'. This is a key distinction. Just as you have to capitalise 'John' but mustn't capitalise 'my brother', you have to capitalise 'Mom' but mustn't capitalise 'my mom'.
Capitalisation (or lack thereof) is not an optional nicety. There are rules.