Not all plants are safe for your birds to chew on. The dangerous plants page
shows some of the common ones but here are a list of some of the safe ones.
Though don't expect your bird to leave much if they get hold of them.
Acacia
African violet
Aloe
Baby's Tears
Bamboo
Begonia
Bougainvillea
Chickweed
Christmas cactus
Cissus (Kangaroo Vine)
Dandelion
Dracaena Varieties
Ferns (Asparagus, bird's nest, maidenhair, Boston)
Figs (creeping, rubber, fiddle leaf, laurel leaf, weeping)
Monkey Plant
Mother-in-laws-tongue
Nasturtium
Natal Plant
Norfolk Island pine
Palms (areca, date, fan, lady, parlour, howeia, kentia, Phoenix, sago).
Peperomia
Petunia
Pittosporum
Pothos
Prayer plant
Purple passion
Schefflera
Sensitive plant
Spider plant
Swedish Ivy
Thistle
Wandering Jew
White Clover
Zebra Plant