What should I use to build a parrilla? Basically, I just need the grate. From what material should it be made, to make it food safe?
Any idea how I can obtain it in a reasonable price?

Per Wikipedia (emphasis added):
Asado (Spanish: [aˈsaðo], Brazilian Portuguese: [assado]) is a term used both for a range of barbecue techniques and the social event of having or attending a barbecue[1] in Argentina (where it is considered the national dish),[2] Brazil, Chile, Colombia, Paraguay and Uruguay. In these countries, asado is a traditional dish and also the standard word for "barbecue". An asado usually consists of beef alongside various other meats, which are cooked on a grill, called a parrilla, or an open fire.
What I would do: 

Buy a grate that is mean for a grill that is the size you are making your parrilla. Usually you can find square cast iron grates for not too expensive and if you need a bigger surface, design your grill in multiplies of the size grate you choose. 

You haven't mentioned the brick work, but make sure you use fire bricks for anything that will be exposed to the heat of the grill.