Yes — peach flesh is safe and delicious to parrots, but the pit contains cyanogenic compounds and must go.
Keeper's noteSoft ripe peach is a great food for converting picky eaters and for older birds. Just never leave the pit in the cage as a 'toy'.
What to watch out for with peaches
Peaches, nectarines, plums, and apricots are all stone fruits: soft, safe, vitamin-rich flesh around a pit whose kernel contains amygdalin, the same cyanide-releasing compound as in apple seeds. Large parrots can and will crack pits open.
How to serve it
Remove the pit entirely before serving.
Wash well or peel — stone fruits carry surface pesticide residue.
Slightly underripe is less messy; fully ripe is more popular. Know your bird.