Isn't Persist Spell a...?

[Darthteej]

Deity only feat? Why are PCs allowed to take it?

[Arundel]

There are, according to the wizards site, 5 sources for persistent spell.

Persistent Spell CAr 81 You can make a spell last all day.
Persistent Spell DD 51 The deity makes one of its spells last all day.
Persistent Spell FRCS 37 You make one of your spells last all day.
Persistent Spell PG 42 You can make a spell last all day.
Persistent Spell TB 41 You make one of your spells last all day.

Source (http://www.wizards.com/default.asp?x=dnd/lists/feats)

Here is a reprint of the CArc entry.
PERSISTENT SPELL [METAMAGIC]
You can make a spell last all day.
Prerequisite: Extend Spell.
Benefit: Spells with a fixed or personal range can have
their duration increased 10 24 hours. Spells of instantaneous
duration cannot be affected by this feat, nor can spells whose
effects are discharged. You don't need to maintain concentration
on persistent detect spells (such as detect magic or detect
thoughts) for you to be aware of the mere presence or absence
of the subject detected, but gaining additional information
requires concentration as normal.
A persistent spell uses up a spell slot six levels higher than
the spell's actual level.

So no, this is a totally legit player feat.

[Yuki Akuma]

Deity only feat? Why are PCs allowed to take it?

No.

Nowhere in the text description for the feat, even as presented in Deities and Demigods, does it limit it to gods only.

And the version found in Complete Arcane, the one everyone uses, doesn't mention deities at all.