Yes — fresh black, red, and white currants are safe and packed with vitamin C and antioxidants. (Note: 'dried currants' are tiny raisins — treat those as raisins.)
Keeper's noteFresh currants are a genuinely healthy berry. The strings of berries make natural foraging. For dried 'currants', see the raisins entry (sugar caution).
Why currants is good for parrots
Fresh currants are non-toxic, low in sugar for a berry, and exceptionally high in vitamin C and antioxidants — blackcurrants especially. They're a healthy small berry for foraging. One naming note: the 'dried currants' sold for baking are actually dried small grapes (raisins), so apply the raisin rules to those.
How to serve it
Fresh currants, washed; offer on the strig for foraging.
Fresh or frozen-then-thawed.
Treat dried 'currants' (raisins) per the raisin entry.