The app fetches it
One file, so it can replace itself. It checks once a day, tells you when there is something newer, and downloads only after you press the button. Nothing arrives unasked.
How updates reach you.
Three ways to install, and updates arrive differently in each. You do not have to choose — you already did, when you picked a file.
.AppImage
.exe
The app fetches it
Same as above, and it downloads only the parts that changed rather than the whole installer. Because it installs per user, no administrator password is needed to update.
.deb
Either, your choice
On its own it updates itself through dpkg, which asks
for your password the way any install does. Add the repository and
your system takes over instead — see below.
Letting apt do it instead.
Add the package repository once and Parallax joins your normal updates. One button in the software centre brings it up to date along with everything else, and the app stops asking.
The app notices when you have done this and steps back — two updaters chasing the same package would offer you the same version twice.
Turning the checking off.
Settings → About → Check automatically. With it off, nothing contacts GitHub unless you press the button yourself.
An update check is a request to a server, which means an IP address and a version number leave your machine. That is worth being able to decline, so you can.