Professor Kukui Ruling

[Zentrility]

Would the card increase Galvantula's Double Thread attack damage? I'm asking because Parallel City reduces its damage through a card effect, so I think Kukui would also effect its damage too. Does it increase damage on similar attacks too from other Pokémon (where there is no number value next to the attack, but damage is dealt through the effect text)?

 

[PokePop]

Well, first thing is to note that Double Thread has had an errata issued:

Double Thread (Galvantula - Steam Siege)

* Due to an error in card text, Galvantula's "Double Thread" attack has been changed. The original text says, "This attack does 30 damage to 2 of your opponent's Pokémon. Also apply Weakness and Resistance for Benched Pokémon". The text has been changed to match the original intention of the attack, which is to affect only Benched Pokémon. Double Thread is updated to say, "This attack does 30 damage to 2 of your opponent's Benched Pokémon. Apply Weakness and Resistance". Effective immediately, Galvantula's "Double Thread" attack will use this updated attack text instead of the original text. (Aug 8, 2016 Pokemon Organized Play News)

If you are not aware, you can find all Pokemon card errata, even old ones, listed in the Compendium.
Here is a link to the errata section:

http://compendium.pokegym.net/compendium-bw.html#c2

Now, regarding your question, since Professor Kukui only adds to damage done to the Active Pokemon and Double Thread, per the errata, only does damage to benched Pokemon then, based on that, Prof Kukui will not add any damage to that attack.

However, IF the card did not have an errata, and IF you had chosen the Active Pokemon as one of the two Pokemon, then Professor Kukui would have added to the damage done to that Active Pokemon.

So, your last comment/question would be correct that damage is damage. It doesn't matter if it is from the big number over to the right or from attack text, if it is damage (and not "place damage counters") then effects can add to that damage.