What adds lost health?

When you interact with a health cabinet you pick up an adrenaline shot which will recover 2 bars of your health when used.

To use the shot, hold up on the D-pad on PS4 and I assume it's the same for Xbone. As noted in the Zozo's answer, the default key is 'C' on PC.

You can use these shots at anytime, usually to recover health during combat.

Edited to add: There is a first aid icon (small cross) in the bottom left hand corner of the screen which shows how many how many adrenaline shots you currently have.

Edited again: You can upgrade your shot to fully recover your health by getting Vito's income to $320,000.

"I've noticed that when you take a medical cabinet you don't get any parts back". You are... half true here.

Indeed when you go to a cabinet you don't get health back, but you get an adrenaline shot. You can use that to replace health (C on pc by default and Colin provided an answer for PS4).

Also you can improve on Vito relation to get additional health bars.

Sidenote: you can add armor on top of it... apparently it absorbs a huge ammount of dmg and for some reason a car crash deducts from your armor before reducing your health (this is a bit of a glitch imo).