(I realise this is an old comment - I'm leaving this here for future learners who might have the same question.)
I've heard both used, but to be honest, "daid" seems to be used far more rarely than "daidí". ("daid" is uncommon enough in my experience to throw me a little when I do hear it.)
Because "daidí" is the Irish for "dad," it just doesn't carry the same juvenile connotation as the word "daddy" does in English... "daddy" and "daidí" are two similar sounding words from two different languages that carry similar but not identical meanings.
In my experience, native Irish speakers (and many Irish born people who don't speak Irish, but whose English has been influenced by these kinds of Irish loan words) just don't trip over this one - I've had 70 year old men tell me in English, "We'll be away to daidí's over Christmas" without a hint of a blush, and I didn't think they were saying "I'm going to my daddy's, because I am five" - I implicitly understood their meaning as "We're going to (Irish loan word alert!) dad's over Christmas."
The same applies to "mamaí,"... I hear "I've got to phone mam" or "I've got to phone mamaí," but in neither case does an Irish adult mean "mammy/mommy."
I think this confuses people who are less familiar with "mamaí" and "daidí" as Irish words in their own right because if you read a lot of content written in Hiberno-English, the Anglicised spellings of "mammy" and "daddy" are not uncommon... however, this is almost always a case of a phonetic rendering of the Irish word, not intended as a diminutive (unless used by children).
To put it simply, the connotation for both "dad" and "daidí" is "my father" (informal) not "my father" (I'm a toddler.)
Á doesn't mean "his" or "her".
Whether a means "his" or "her" or "their" depends on what happens to the following word:
a chat - lenition - "his cat"
a cat - no change - "her cat"
a gcat - eclipsis - "their cat"
You can't lenite or eclipse words that start with a vowel, so the rules are a bit different:
a athair - no change - "his father"
a hathair - h-prefix - "her father"
a n-athair - n-prefix - "their father"
Words that start with letters that can't be eclipsed or lenited (like leabhar) need context to tell whether a means "his", "her", or "their".