Leave the new parts in their original boxes and store the boxes in plastic bins to keep them safe and clean. You don't want to lose the part numbers or other manufacturer info. Also put the date purchased. Save the receipts in a folder for each car so if you have to reorder or the part is defective or wrong, you can return it. Many suppliers won't take back stuff not in the original packaging or dirty packaging. I once lived at a house with a one car garage (14x20). I had stored 5 complete disassembled engines with extra parts, had Monte Carlo body parts in the garage attic, and still had space for the car, mower, bicycles, workbench, tool cabinet and air compressor so organization CAN be done with creativity.
I bought several older file cabinets to store various items, such as polishes and waxes, rags and towels, and things that need to be store vertically. The file cabinets are fairly narrow and can fit in spots that are too narrow for wider cabinets (I also have several of these). The file cabinets were purchased at local thrift stores for about $20 each. Each drawer is labeled on the outside, and the pull-out sliding drawers make it easy to find things.
I also like the horizontal magnetic strips for attaching hand-tools to the inside of the built-in wood/formica cabinet doors. Available at Harbor Freight.