As far as I know, on a 2006 Odyssey, the OEM Honda Remote will not lock/unlock the doors while the engine is running.
It didn't do this before you added the remote start, and it will not after.
(I am sure somebody could make it work somehow)
Neither will it start the engine with the OEM 3X Lock.
Not sure the OEM Honda Remote will open the sliding doors while in park, as you say. I have not tried it.
I also have an '06 Odyssey.
I personally use an EVO-ONE with a RF642W remote.
The RF642W remote does everything (as far as I know) the original remote did, so I no longer even carry the Honda Remote at all.
Remote Start, Lock, Unlock, Open Either Sliding Doors,
Someone told me I can even open/close the windows, I honestly have never even tried.
I would ***think*** you need something other than a 1 Button Remote.
I'm pretty sure you have many options to add a multi-button, 1-way remote to your current system.
An expert will answer shortly and I'm sure will give you correct options to upgrade your 1 Button Crimestopper remote.
If a cheap upgrade exists that will start, stop, lock, unlock, panic, and open the sliding doors, (maybe even roll up/down the windows too)
I would suggest you get that and ditch the OEM Remotes alltogether.