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I've had a fierce discussion with a friend regarding the Arcanist.

He insists that for everything he counts as both a sorcerer and a wizard, in regards to qualifying for it. Feats, etc.

I am hesitant to agree. Perhaps it's true for feats, but an arcane discovery is not a feat, and it is strictly a wizard thing. Plus, the Exploit to get one is a bit superfluous if one can just spend a feat and use one's full Arcanist level.

Everywhere I look, people seem to agree that if an Arcanist wants to learn an arcane discovery, they have to do so through the exploit.

My argument has been along the lines of the answer given here Is an arcane discovery a feat?

So what I'd really like to know is: Supposing the link gets it right. Is there somewhere that says, precisely, in which regard the hybrid classes (especially the Arcanist) counts as its parent classes?

What your friend describes was discarded during the Advanced Class Guide playtest (found here, more specifically on this developer post). Those hybrid classes would not only count as both parent classes, but would also lock you out of those classes for multiclassing purposes. For example, a Skald would count both as a Bard and as a Barbarian, and you couldn't multiclass Skald/Bard or Skald/Barbarian at all.

That rule was discarded in the final version of the hybrid classes

And here are the final rules for Parent Classes:

Parent Classes: Each one of the following classes lists two classes that it draws upon to form the basis of its theme. While a character can multiclass with these parent classes, this usually results in redundant abilities. Such abilities don’t stack unless specified. If a class feature allows the character to make a one-time choice (such as a bloodline), that choice must match similar choices made by the parent classes and vice-versa (such as selecting the same bloodline). The new classes presented here are all hybrids of two existing core or base classes.

Now, they are their own thing, you can multiclass just fine, and you do not count as being a member of the parent class unless you have a special class ability saying that you do. For example, the Brawler's Martial Training says:

a brawler counts her total brawler levels as both fighter levels and monk levels for the purpose of qualifying for feats. She also counts as both a fighter and a monk for feats and magic items that have different effects based on whether the character has levels in those classes (such as Stunning Fist and a monk’s robe).

The Arcanist lacks an ability like this, and thus does not count as wizard and/or sorcerer for feats, magic items, prestige classes, or anything that depends on you being a member of either of the two classes.

Regardless, arcane discoveries are not feats. But an arcanist may take the Extra Exploit feat to learn additional exploits.