When I front squat I have no problem going ass to grass but for back squats I can barely hit parallel.

If I want to hit great depth on my back squats I throw a plate under my heels before squatting and that's a temporary fix until I increase my ankle and hip mobility.

Will front squatting more help increase my mobility for back squats?

EDIT: Found this from https://www.t-nation.com/training/fix-your-front-squat that explains why I can get lower on my front squats compared to back squats

When a barbell is loaded on the front of the body, the pelvis gets to tilt backwards somewhat, which makes the hamstrings less taut. This gives them the freedom to allow a greater ROM at the bottom of the lift. This pelvic tilt also allows the lower abs to contribute to the lift more, and takes the hip flexors away from "blocking" the movement.