The short answer

Ripe flesh only, in tiny amounts — and never the green parts.

Why

Ripe tomato flesh is safe in small amounts, but it's quite acidic and can upset a budgie's crop. More importantly, the leaves, stem and any unripe green tomato contain solanine, which is toxic, so those must never be offered.

How to feed it

Offer a tiny piece of washed, fully ripe flesh only — no leaves, stem, or green tomato.

How much & how often

A small amount, occasionally.