In English, "dad turns" is a slang term for a style of skiing. If you are making "dad turns", you are skiing like an old man.
I found 親父 which I believe is "pop". Is that the best translation for "dad" in this case?
I'm struggling to find the translation for "turns", the plural noun, as in ski turns.
Please read my comments under your original question first.
In this post, I'm specifically answering only to:
I'm struggling to find the translation for "turns", the plural noun, as in ski turns.
According to this page スキー用語集 that compiles a list of terms related to skiing, a turn in skiing is simply:
ターン{taan}
which is the transliteration of the English word to the Japanese writing and phonetic system.
This is the definition for ターン{taan} provided in this site:
which translates as
To rotate and change your direction.
Turn is ターン. Period.
As for Dad, おやじ (オヤジ, 親父) is not bad, but you may want to choose おっさん or おじさん instead. おやじ can be used to carry a connotation of "dandy" sometimes, and recently it may even appear on the cover of a fashion magazine:
On the other hand, おっさん is often used to playfully express something that has an "old man vibe", so if you want to humorously convey a dad-like action, おっさんターン or おじさんターン might be more fitting. If used appropriately, it wouldn't sound too negative.
Note that this is just a literal translation attempt. With おっさんターン, even people who are not familiar with skiing can infer that it refers to some traditional-yet-uncool turn, but those who are knowledgeable about skiing may not necessarily understand what specific style is being referred to.
パパターン, 父親ターン and so on are simply confusing.
For another example, this writing style featuring lots of emoji is often referred to as おじさん構文 (but not オヤジ構文):
