Do you write mum and dad with a capital letter at the start for English?

You capitalize mom and dad when you use it as their name. For example - "Hey Mom/Dad, can I have a ride? It's not when you say, "My mom is cool."

The rule is basically this.... If you can insert their given name, Joan or David, for example into the sentence, you capitalize it, because you would capitalize their given name if it were used. If you cannot use (properly) their given name in the sentence, you do not capitalize mom or dad. For example: "My mom and my dad grounded me for a week because I was bad." In this instance you would not capitalize mom and dad because inserting their given names, say Joan and David, would not be correct. For example: "My Joan and my David grounded me for a week because I was bad," is not a correct sentence and thus when using mom and dad in that instance you would not capitalize them. You would capitalize mom and dad in the following sentence because you can use their given names properly and would capitalize them so you would capitalize mom and dad. "One time, at band camp, Mom and Dad came by to hear us play." You can use Joan and David properly in this sentence, so you would capitalize Mom and Dad.
If you are referring to your mom in a generic term and are saying "my mom," "his mom," "their mom," etc, no. If you are referring to her by Mom being her name, yes.

Example: Today, my mom picked me up from school.

Example: Today, Mom picked me up from school.