Rayleigh is Imu's descendant and Imu isn't a parrot, he is...
I tried to make a connection between Imu and Rayleigh (coz I believe Ray is Mihawk's dad) and I think I struck gold.
I found it strange that Imu, Joyboy's buddy, is the literal Devil and our Pirate King's first mate is coincidentally called King of the Underworld. Two similar figures (Pirate Kings) with friends associated with hellish themes.
So being a firm believer of a Imu parrot theory I typed "rayleigh bird" into google to find what bird could Rayleigh be based on and this little guy popped up

Say hello to Imu.
This dude is called a Lyrebird. I know Oda laughed his ass off when he first heard of this bird, this thing comes with a built-in pun! Its a Liar bird!
Here is why I believe Imu is this bird
1. Sixteen feathers
Yup, you read that right. This bird's most prominent feature, it's tail, is made up of 16 feathers.
The same 16 as in Doflamingo's 16 Holy Bullets
The same 16 as in Mihawk's 16 Day Moon
The same 16 as in the 16 bullets in Donquixote Mjosgard (rip)
The lyrebird is the only bird with a direct connection to number 16
2. Foot
These birds have a particular foot arrangement called anisodactyl arrangement. In this arrangement, 3 toes face forward and one toe faces back.

Certainly reminds me of something

3. Mimicry
When it comes to mimicry these birds leave parrots in the dirt. These are nature's best mimics. I looked it up on youtube and these guys can replicate the sound of ringtones, car alarms, shotguns, fucking chainsaws and human voices perfectly. It can mimic over 20 different bird species (yup I locked in too when I saw the number 20)
Maybe Imu's actual voice doesn't sound like the South bird and it's just mimicking it.
And Imu has been doing the same since he came to Elbaph. He is a natural copycat creature with powers similar to our MC.

4. Name
A lyrebird is a songbird. It's named after a lyre because the feathers on it's tail look like a lyre. For those who don't know (I didn't until I researched this bird) a lyre is musical instrument. Its like a little harp. They come with 16 strings too wink wink

Oh and because Imu is a liar. The world government is built on lies and Imu is desperately trying to hide it. Just like his counterpart Doflamingo did.
5. Genus name
Its genus name is Menura which is a combination of Greek words for 'moon' and 'tail'. They are named as such because of the crescent moon patterns on their tails.
Who better to oppose the Sun God than the Moon Devil? or something like that.
But wait, aren't we forgetting something? What about Rayleigh? Why did googling 'rayleigh bird' give me the lyrebird? Rayleigh and lyrebird don't sound the same so what's up with that?
So lyrebirds are exclusive to Australia. And there's this place in Australia that is famous for it's bird fossils. The name of the place is

Riversleigh
Silvers Rayleigh
Ladies and gentlemen please give a huge round of applause to our boy Eiichiro Oda
But stop right there I got a little more! In chapter 1161 when Rayleigh moves in to save Shakky from Sommers he uses a named attack, Higanbana

Which means red spider lily. And another translation for it is
Billhook of the Beyond
Bill as in a bird's bill
Rayleigh is a human based on a bird and has bird-like behavior similar to Doffy, Mihawk etc
Oda has done a great job of hiding Rayleigh's birdy nature in plain sight. He has given Rayleigh a task that many birds do but most people don't know about. And this is the first piece of lore that we find about him.
Waterproof Coating

Our boy Rayleigh coats ships with a waterproof resin so that they can travel safely underwater.
Waterbirds (birds that live around water and hunt fish) have an organ called the uropygial gland. What does it do?
This gland secrets an oil that birds use to coat their feathers. It makes them waterproof and let's them dive into the water to hunt fish and fly away with ease.
Making this connection should have been enough for me, but it wasn't.
I couldn't rest until I found out the exact bird that Rayleigh is. So I searched and searched and searched and searched...
Until I found it.
Ladies and gentlemen, I present to you
The Silvery Kingfisher

A bird that coats its feathers, hunts underwater fish and has 'king' in its name.
Riversleigh + Silvery Kingfisher = Silvers Rayleigh, King of the Underworld