Capitalization is required for proper nouns, such as names, but sometimes words can be proper nouns or common nouns. Family titles, such as mom and dad, fit into this category.
The general rule is to capitalize a family name when it is used as a name, and not to capitalize when it is a common noun. A good tip is to look if there is a pronoun or article (e.g., the, your, his, etc.) preceding the title. If an article or pronoun is there, don’t capitalize (e.g., your mother is pretty). If there is no article or pronoun, then the word is being used as a proper noun and is capitalized (e.g., that dress looks good on Mom).
Side note: Be aware that when you are using a family name as a proper noun in dialogue, sometimes a comma is required as well.
Frye’s father, Mark, experienced chest pains and suffered a heart attack at the stadium just before halftime. His mother, Dana, let Ian know what was going on. Ian went into the stands to check on his dad, who was being attended to by emergency medical personnel. [Deseret News]
For a while, neither Dad nor I felt much like picking up the phone. Then, two years after we lost Mom, I was home for Christmas and saw Match.com installed on Dad’s phone. [Huffington Post]
For Weaver, it “tidies up a few of the loose pieces on the family tree,” and his daughter says she “thanks Dad for her high cheekbones.” [Guelph Mercury]
Thanks, Mom and Dad. [Post Independent]