There are two ways to do this. The first one is a bit complicated and the instructions while written for Kali also works for Parrot.
The second way is to use VirtualBox to install Parrot directly onto the USB drive as if installing to a hard disk. Just make sure to install the MBR to the USB drive.
I haven’t tried it, but I assume that like Kali’s live encryption option when boot off of is the same as how Parrot’s work.
The first method allows you the option to boot into the live CD without persistence (just RAM). As of a few Kali versions ago, the Live ISO does not have an installer as they have separated the Live and Install into two different ISOs. I believe booting into Parrot install using the Kali link’s method would allow the USB to be a Live installer and/or a persistent install.
My second method using VirtualBox just makes a bootable, encrypted, fully persistent install. When you boot off the USB, it will first ask for the encryption password before showing the Grub2 menu to boot into the (updatable) full distro.
Not that I know of. I’d do method #1 to allow either boot into default Live mode or encrypted persistence mode at the Grub menu.
I tried my method of installing Kali 2022.3 onto a thumb drive (using VirtualBox) and it didn’t work. I’ll try using an older Kali build later. (I’m near Tampa which is about to be hit by a hurricane = no power for possibly weeks.)