You're almost certainly running out of space the Security branch of Parrot OS. You should try home or perhaps fall back to when they released a mini and dist-upgrade from there (might be wonky, there was a bit of silliness between versions 3 and 4 involving tomoyo but you're mostly fine if you just type in what it asks and continue to apt update && apt dist-upgrade
). You really should give that sucker more like 60GB.
I'm pretty sure a x86_64 security version rolls out to a whopping 30GB once you run each tool once or so (many of them have large databases that need to be populated looking at you OpenVAS/metasploit/etc..).
You're on the right track listing it as debian.
Another Pro-tip: Always run the installer from the first prompt, don't jump into live and then run it from there. I'm not sure that's ever worked for anyone except palurino himself (if that). So, if you're not going live or persistent live, definitely just go straight into the installer.
Your best bet is likely the current "Home" release, then just cherry pick your tools from there.
You'll definitely want to verify your ISO image download too, since that's one of the most common reasons anyone's installation of anything fails (corrupted image).
sha256sum /path/to/iso
It can happen to anyone.
1Try to use older/newer OS version. Add the full error message. – Biswapriyo – 2018-09-16T17:44:34.647