GP Parrrley does not proc ludens

That's intended. It only procs on-hit effects as a melee attack (Sheen, Hydra etc.) Ezreal's Q doesn't proc it either. However, with Runeglaive it will because of the damage it does.

This is intended. Parrrley is treated as an auto attack when it comes to on-hit effects. This means that it will only proc anything that auto attacks proc, and since auto attacks do not proc Luden's Echo, neither will Parrrley.

I think it's because it is considered more of an auto-attack than a spell, much like Ezreal's Q.

I'd bet GP's ult procs Luden's.

It's an enhanced auto attack, which is why it procs Shiv and Hydra. It's not classed as a spell, which is required to proc Luden's

It shouldn't anyway.

http://i.ytimg.com/vi/nm4Qb6UqpNc/hqdefault.jpg