I understand that they have similar meanings, but would it sound awkward if I say "mother" and "father" instead?

Native speaker’s answer
Rebecca
"Mother" and "Father" are more formal in tone, so it's rather rare to hear people use these forms of address today, especially amongst young people. You might hear older people using this, or when people are referring to their parents in rather formal situations. Children usually call their parents "Mum/Mom/Mommy/Mummy" and "Dad/Daddy." Ex: My mom and dad have been married for twenty years. Ex: Mommy! Can I have a bedtime snack?