*cracks knuckles* okay take a seat
Apollo being a good parent is not a surprise when you learn that he is the nurturer of young, originator of education, and the primary god who presides over the transition of youths into adults. But here are the reasons why he deserves the title "The Best Dad" :
He takes care of his children and often educates them personally
He also gifts lyre to Orpheus and teaches him to play it, trains Linus to play the lyre as well. Note that all these kids were not even divine, just mortal kids. Like most other gods he could have just brushed off the responsibility, or put it onto some nymphs. But he didn't abandon them and played a crucial role in their growing up.
Not only his own children, he has acted as a foster father to Chiron, who was abandoned by his mother. Then, he also adopted Carnus, a son of Zeus, and raised him with the help of Leto. He might as well be the god of adoption.
He is constantly involved in their life
Apollo keeps showing up in his kids' life now and then. Like, look this wholesome content of Apollo being dadpollo for his son Aristaeus. When Ari is leaving for the war, Apollo leaves all his work behind to come and suit up his son. And later, he also goes a long distance to save him.
Once Aristaeus and Dionysus contest to see which is better, honey or wine. All the gods prefer wine so Dio wins, but Apollo is grumpy that his son lost.
Wine is dear to Apollo. He even has an epithet which translates to "heavily drunk". And yet when his son lost, he is not happy and he's like "Wine?! I hate wine! I can't believe people love it ugh" XD It's like that wholesome moment when parent support the child with something even though they don't personally agree with it XD
Also, here have a look at Apollo being a good wingman.
His daughter Phemonoe is said to have been his Pythia, so there definitely was a lot to father-daughter bonding.
Apollo is emotionally attached to his kids.
There are many instances of Apollo losing his mind over his children's death, it is like a defining trait of his. The most famous example is how he dealt with Asclepius' death. Apollo literally cried a river over his son's death. Even Zeus felt sorry for him and brought the mortal back to life when requested.
Apollo the Muses together mourn for the death of their sons (Orpheus and Linus resp)
Here again Apollo cries when his son Amphiaraus is about to die.
It's touching, how he is such a powerful god but breaks down sobbing at the death of his children. Gods rarely display such emotions. I have not seen anyone other than Ares and Apollo actually cry when their children die. Not just mourning, he also seeks revenge on people who cause harm to his children. He kills the cyclopes who had designed Zeus' thunderbolt with which Asclepius was killed. On two separate instances, he sends plague to villagers who killed his sons (Linus and Carnus). One of the reasons Apollo was bent on killing Achilles was because Achilles had killed 2 of his sons. (Troilus and Tenes) (And in fair number of traditions Hector was Apollo's son as well)
After they die, He creates something in their memory:
So overall, his children mean A LOT to Apollo. He loves them, cares for them, helps them. He has the epithets "Patroos" and "Patir" which mean "lord of fathers", "fatherly". In Athens, legal adoptions were made on Apollo's birthday. So Apollo is 💯/💯 dad material. You're lucky if you're a child of Apollo, you've got an awesome dad and you don't have to be a great warrior in order to be famous and respected.
(Bottom note: We have to remember that Apollo is a god. One of the busiest gods even. So he can't fill the role of a father like a human does. But, he is better than his fellow gods. I'm not saying he is the only one who cared. Ares and Hermes are next on the list, and Olympians in general had soft spot for babies, but Apollo is just on a different level)
(all the screenshots of literary sources are taken from theoi.com and wikipedia)