No you can't. Possession of all the parts to convert a firearm into a full-auto firearm are themselves classified as a fully automatic firearm. They require the same paperwork and tax procedure that purchasing a fully automatic firearm does, and that procedure (for non-government agencies) was essentially abolished by an amendment to the McClure-Volkmer act in 1986.
You are confusing spare parts kits with conversion kits, or you are talking about illegal products (such as drop-in autosears, possession of which without the tax and paperwork is a felony).
Correcting myself; the procedure for new manufacture (post 1986) was banned. It is still legal to purchase full auto firearms (and conversion kits) manufactured before that date (although those too must go through the paperwork process and payment of a $200.00 tax).