Yes, you need sunscreen on cloudy days.

Clouds block visible light, not UV rays — especially UVA, which causes aging and skin damage.

Up to 80% of UV rays penetrate cloud cover. Rain doesn’t help either. If you’re outside for more than 15 minutes and care about your skin, put it on. The only exception is if you’re in a basement without windows.

A cloudy day just means you skip the reapplication every 2 hours, not that you skip it entirely.