With Reforging abilities added to the Tower Gunsmith we now need Weapon Parts more than ever. It takes 1 Mote of light, 250 Glimmer, and 3 Weapon Parts to re-roll a weapon. Below are some of the best and fastest ways I have found to Farm Weapon Parts.
Get 7 Wins in Trials of Osiris and buy the weapon Brother Vance is selling, break it down and it will yield anywhere from 2-4 weapon parts. This is the fastest way to get weapon parts but it is also the most difficult as you will need a team that can get 7 wins in Trials. Then you just farm glimmer and keep buying the weapon and breaking it down. I made 500 weapon parts last week using this method… it sucked and took a lot of time but it is the fastest way to get parts.
Buy green weapons from the Gunsmith and equip them when you turn in bounties. This is probably the easiest way to get parts, breaking down a Green or Blue Weapon that has its perks unlocked will generally yield 2-5 weapon parts. I do this everytime I turn in bounties and it makes me anywhere from 50-100 weapon parts per day. However it is very expensive and can drain your glimmer real quick.
Do Strikes and break down any blue or green weapon engrams you get. Pretty self explanatory, this yields the least amount of parts but at least you get something.