Put all the parts that match one particular model of machine in together, bundle them on pallets, properly secured and wrapped for weather protection - then provide an accurate parts list showing what is on that pallet (part #'s and qty).
Take good photos of the parts prior to packaging, then put each pallet up for "offers" on eBay as a "bulk, wholesale lot". eBay gives you the widest exposure to potential buyers. The best offers should come from companies specializing in NOS parts and components. You may get satisfactory offers from small "wheeler-dealers".
As a guide, you can generally expect to get around 20%-25% of retail value, and the buyer normally sells at 40%-50% of retail value. Be aware that most people place the retail value of a part or component on the cheapest aftermarket source price for that part/component.
So, if the Cat price for one particular part is $500, but a potential buyer can acquire it in a satisfactory quality aftermarket deal for $250, then $250 is the "retail value", even though you're selling genuine Cat parts/components. Cat retail prices have large inbuilt margins for fat Caterpillar profits, and Cat research outgoings, and Cat senior employee perks and benefits.
The alternative to selling on eBay is the used equipment sites such as RockandDirt and Machinery Trader. The costs of selling on those sites is less than eBay, but the total market exposure is lower.
Remember that many parts change in pricing over the years, so find out current pricing. Also, engineering changes often make some parts/components useless, as they become obsolete due to the engineering change to a new design.
In addition, many potential buyers will purchase a different part or component to the original, if it's a part/component that will do the same job at lower cost. Typically, this is items such as external rear view mirrors, where an OEM mirror may be $300, but an alternative mirror can be fitted, that only costs $100.