Yes — mango flesh is an excellent source of vitamin A, which many parrots lack. Peel it first: the skin contains urushiol, the poison-ivy compound.
What it means for budgies
Keep mango to a small treat for budgies — they don't need much sugar, and they're prone to obesity and fatty tumours. A budgie portion is tiny — a single bite or a little grated amount is plenty.
The full picture on mango
Vitamin A deficiency is one of the most common nutritional problems in pet parrots, and mango is one of the tastiest ways to fix it. The flesh is entirely safe; the skin contains urushiol, which can irritate sensitive birds (and people), and the pit is a slippery choking hazard.
Always peel; discard the skin and pit.
Cubes, spears, or mashed for chicks — all work.
Plain dried mango (no sugar, no sulfites) makes a good travel treat.