Just to clear up some things instead of making this binary/black and white: There is a time and place for quarter and "full" (just below parallel, not "ass to grass") squats.
"Full" squats are for overall strength - they hit the most muscle mass over the longest effective range of motion (ROM). You want to overall build strength to make your body as resilient/strong as possible while developing your muscle's ability to produce force.
Quarter squats have shown to be beneficial for improving spring times since the ROM is more sport specific. But be warned you risk building more muscle imbalances as they are heavily quad dominant and hardly hit the hamstrings while really loading up the knees.
Just like doing standard running training, periodization is key. Before you start trying to hit those heavy quarter squats (especially for novice lifters), do a block of regular full ROM squats to build full body strength, then incorporate half squats - to make it easy, switch off between every session. Don't let ego lifting brain take over by jumping into doing quarter squats first as you chase "big" numbers (no serious lifter is going to be remotely impressed or take you seriously by your 200 lbs. quarter squat) and opting for the easier lift (which, frankly, it is).