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Replying to and
The my is implied so why would you say it? Your dad or Billy's dad is needed because it is needed to specify who you are talking about. "Got car washed today" implies that it is my car without having 'my' in front (assuming there isn't a predetermined context about the car).
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Replying to and
Whoa. First time I’ve encountered this question outside of the comically long-standing bit of debate amongst myself and a couple of my friends. I’ve given it a lot of thought.
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Replying to and
Without the "my", these words are used as names. They would be the same names that your friends call their parents as well. This context may help explain the usage. Keep in mind that you would capitalize Mom and Dad when used as names when writing.