Firefly - getting rid of parts (dumping cargo)?
From the official FAQ,

Question: How do I get rid of stuff on my ship I don’t need or want anymore?
Answer: On your turn, you may dump Fuel, Parts, Cargo or Contraband at any time. You may drop Fugitives and Passengers off in any sector with a Planet. You can’t toss them out the air-lock in deep space! This does not use an Action and can be done during another Action, on your turn.

You also have the option to trade with other players. From the rules,

When two or more players’ ships are stopped in the same Sector, players may buy, sell and trade Crew, Fuel, Parts, Cargo, Contraband, Ship Upgrades and Gear to one another without restriction. Players are free to discuss and make these deals outside of the normal turn order and other players may take their normal turn while bargains are struck. Dealing with others players in this fashion does not require using an Action. The opportunity for Trading ends when one of the players leaves the sector.

There doesn't seem to be any option of selling back to the house, but we house-ruled they can be sold off at half-price at Suppliers. This has allowed people that would otherwise have been eliminated from the game to keep on playing.