Q:

Mom and Dad as a phrase; Do we need articles in "I have (a) Mum and (a) Dad..."? 

I am a bit confused with a correct usage of Mom and Dad and articles.  
My daughter wrote a short essay starting from:
"Hello my name is ... and I am going to tell you about me.  I have a Mum and Dad..."
At first she used "Mum" as she always does in school.  I found that the word mum is more about sound not a person and it should be Mom instead. 
Secondly she use "a" to "Mum and Dad".  Shouldn't be "a Mum and a Dad" or without an article at all? 

A:

Mom vs Mum is an American vs British question.  Most would understand either way.  Yes, "mum" also means silent, but when used as noun, it's well understood to mean the woman who gave birth. 

I have a Mum and Dad

Is okay; Mum and Dad are a matched set like salt and pepper shakers. ;-) 
Also correct but perhaps slightly more formal:

I have a Mum and a Dad

Omitting the article altogether would just be wrong.  It can really only be omitted with things you measure, rather than count.  E.g.

I have rice

not

I have a rice

Of course, once you have multiples of something you can count, then the article can be left out:

I have a cat
I have cats

