Aviary netting is intended for use with gamebirds and other soft-billed birds. It is not suitable for hookbills, because a determined parrot can chew right through it. It is a woven product, heavy duty and strong, but flexible to protect against head or neck injuries, should a bird fly up and hit the net.
For a parrot aviary, your best bet would be stainless steel hardware cloth. It is metal wire, woven or welded into a mesh grid. Pick a mesh size that matches up with the size and beak strength of your bird. Galvanized hardware cloth and vinyl-coated hardware cloth are also available. These alternatives are cheaper than stainless steel and easier to source locally, but they have some safety concerns.
Galvanized metal is coated in zinc, which is toxic to birds if consumed. This is usually not an issue if you are keeping chickens, finches, canaries, quail and other soft-billed birds. But once again ... hookbills love to chew. So they can potentially expose themselves to heavy metal toxicity, if they chew on the wire mesh. Chewing on vinyl coated wire has the potential to result in plastic consumption. Less risky, but still not a good thing, especially if you have an avid chewer. If you can find it in the right size and afford the price tag, stainless steel is best.
As for wood, avoid using pressure treated or painted woods anywhere the bird can reach with its beak. Exterior areas that can't be accessed by the bird are fine, but any wood on the "inside" of the aviary is going to get chewed. It is entirely possible to design aviary panels so the wire protects the wood from curious beaks. Just plan out the structure carefully and block any exposed wood with chew-resistant materials. For example, external structural elements could be made using treated wood, while the interior features are all made using raw untreated lumber and stainless steel hardware.
Although cedar is naturally weather resistant, it is not a great choice for an aviary. Cedar is an aromatic wood and the strong smell comes from natural phenols that can cause respiratory irritation for birds. At high levels, exposure to aromatic cedar fumes can cause death for small birds. Be careful of cedar closets, cedar cooking planks, and cedar shavings, among other things.
That being said, there are many species of cedar and not all cedar wood has a powerful aroma. Some species are considered safe for use for birdhouses, so I would think they would also be potentially acceptable for an aviary. Also, most aviary designs are quite open, allowing for good ventilation, and over time, the smell dissipates completely. But it is unclear to me how directly toxic cedar oil is to birds. The fumes cause respiratory issues. I don't know if there is any direct danger from chewing on the wood. Cedar shavings are avoided in rats/mice, but again, it is due to respiratory issues from the fumes, not direct toxicity from contact with or consumption of the shavings. Safest option would be to avoid it.
Also, be careful to pick up any pieces of galvanized metal or screws that fall during construction. Even a small piece of metal can cause a bird to get sick, if he eats it.