I’m 36 and in the U.S., if that matters.
When I was growing up, I referred to my parents as “my father” and “my mother.” So did all the other kids. Mom and Dad were their first names, as far as I was concerned.
It seems that people about ten years or so younger than me often use “my mom” and “my dad.”
To my ears using “my dad” to describe the man who raised me makes about as much sense as saying “my Nicole” to describe the woman who cuts my hair.
Is one considered more grammatically correct?
It’s not a grammatical question. There’s no error in grammar in treating “dad” as a noun like “father”.
It’s a question of usage. In many varieties of English - including my own - “dad” is a noun like any other, and “my dad”, “your dad”, “his dad” etc are all unremarkable. Just as your father might address you as “son”, and refer to you as “my son”, so you can do the same with “dad”.
Clearly that wasn’t so in in your particular variety of English. From what you say, either your variety of English is changing, or you have moved to somewhere where the local variety of English does not share this feature with your variety.
I’m 44, in the U.S.
I say “my mom” and “my dad” almost all the time, but I tend to speak in the vernacular. I think “my dad” is just as grammatical as “my father,” just less formal.
You can call your parents Mother and Father, or Mom and Dad. You can refer to them as my mother and father, or my mom and dad.
There’s no issue of grammar here whatsoever. In American English, at least the varieties I’m familiar with, it’s just an issue of formality. “My dad” and “my mom” is more familiar (less formal) than “my father” or “my mother.” They can be used in the vocative (as a term of address) as plain “Mom” or “Dad,” but they are used just as freely as casual substitutes for the more formal “mother” and “father.”
I’m 35, grew up in Chicago. I don’t ever recall anyone ever having an objection to the “my mom” and “my dad/pop” construction.
At first glance I would have agreed with the other posters in thinking that “my dad” vs. “my father” has nothing to do with grammar but rather just with word choice. But upon further reflection, if someone grows up treating “mom” and “dad” exclusively as proper names, as the OP did, then “my mom” and “my dad” should indeed sound grammatically odd, just like “my Nicole”. That being said, asking which one is considered more grammatically correct is problematic, because “grammatically correct” must always be relativized to a judge, and judges are known to differ in their opinions. As an appeal to historical precedence, though, I offer up the fact that according to the OED, the collocation “my dad” is attested as far back as the sixteenth century.
Using “my” with informal names for parents like “dad”, “mom”, “mum” or “pa” seems completely unproblematic to me. Indeed, using “my” with proper names like “Nicole” seems reasonable, especially to distinguish one of two or more people with the same name. Foir example, is there anything wrong with following?
There were two Nicoles in my high school class, and one is now my girlfriend. My Nicole is currently visiting her parents, while the other Nicole is currently at the same party as I am.
I don’t think there’s any problem with that.
“My Dad” sounds perfectly normal to me, although I would capitalize it as well as Father. But then, referring to a cousin as “our Declan” soudns right to me too. . .
I was actually just thinking about how this worked in other languages. I grew up in a Polish household, and called my parents Mama and Tata my whole life. When speaking about them colloquially in Polish, I would say moja mama or mój tata/tato (“my mom,” “my dad”) vs the more formal moja matka and mój ojciec (“my mother,” “my father.”) The difference feels the same as in English and corresponds to the level of formality and diction I want to display in my language.
There’s a pretty good consensus on this, and I think pulykamell expressed it particularly well here.
Also remember that for certain kids (or the adults they grew into), they may be gently making a distinction important to them: “my father” = “he who begat me, my sperm donor” vs. “my Dad/-dy” = “male care-giving parental figure”. This distinction was none too gently drawn for me to the best of his ability by a five-year-old ten years ago, and I’ve never forgotten the searing lesson he taught me that day. His father was almost never around; he was proud of his Daddy who loved him and was there when he needed him.
There’s nothing wrong with that, but it would be a very specialized situation. Generally speaking, American English doesn’t use the “my Nicole” construction to routinely refer to a person. This is different in many British dialects, though, as illustrated by TruCelt –
I would quibble with the capitalization. “My dad” and “my father” because you’re using the words in their common noun senses, not as proper names.
I’m 53, grew up in the medwest and suburban Baltimore, and have always used and heard “my mom” and “my dad” commonly, and sometimes even “Mom and Dad” when talking to someone who knows them: “Dad said I need to get my homework done before I go out.” So it’s not just people 10 years younger than you. 
If you distinguish dad and mom because some people use them as proper names, then what happens in those families in which father and mother are used instead? There are some. Every variation of father, dad, pater, pop, pa, da, daddy and a million others plus their female alternatives are used.
No matter how you treat one, you have to treat all the others the same way. They are entirely equivalent.
Avoid questions by using “male progenitor” or “sire”.
Unless you’re trying to legislate your personal taste, that is just false as a description of how language works. To take a not entirely unrelated case, “doctor” and “physician”, in common parlance, would seem to be more or less synonymous. And yet only the former can be used for direct address:
The moral here is that you can’t deduce from the meaning of a word alone what kinds of grammatical uses it has. And indeed for the OP, “dad” and “father”, while synonymous, have different uses.
Although I agree with you, if the speaker sees dad as a proper noun in itself akin to Nicole, then it makes sense to capitalize it. You wouldn’t write it as “my nicole,” after all.
But that’s what I tell my typists: it’s “Dad” or “my dad.”
Physician, heal thyself.
I was under the impression that one should be careful with capitalization.
for example: My dad had dinner with us last night.
While visiting, Dad told a funny story.
Putting my in front of dad turns it into a general noun. You could just as easily replace it with the word goldfish. But the second sentence only makes sense if you replace Dad with another proper noun.
Am I correct about this?
That’s the rule I follow.
Nope, I am a descriptivist. I’m reporting how the language works. Whether it’s my dad, my father, my da, my mommy, my mammy, my moms, or whatever, these particular usages are entirely equivalent. They are grammatically identical. That’s why - as everybody has correctly said - this is not a grammar issue.
I didn’t say a word about any others. And others are not the same. Because usage declares they are not the same.
Okay, fair enough, but I think it bears emphasizing that you are describing your grammar and how your language works, not how the language works for everyone. If I read the OP correctly, his grammar is different from yours in such a way that there is a grammatical distinction between dad and father. He may be in the minority on this, but that doesn’t change these facts.