It is very easy, click the lab, Air Lab, Army Lab or Sea Lab you will find the pictures of the weapons you want to upgrade, and a button " Search" click the search button of the weapon you want to upgrade,and you will see the parts pictures and 2 buttons asking you; " Ask for parts" or "Buy for...." Buy For.... means you have to use your Empire Points. So if you want to complete the search in seconds then pay the Empire Points if not, click " Ask For Parts". I hope you have understood.
I will add there are 2 forms of "ask for parts", along the lines of other Zynga games.
One type places a request on your wall, with a link that will allow a visitor to click to send 1 item to you, and receive a duplicate item in return.
The other form requires individual selection of your friends to request a given item. Each friend can only be sent one of these type of request (for all items of this sort) per day. If a friend sends this item, only you will receive it - all your friend will get is your thanks.
I don't have a list of which item falls under each category - perhaps something to add......