I expect the forklift came first, for use with the ancient dunnage system. Dunnage (a word with several meanings, actually) is basically a couple of scrap planks strapped, tied or simply placed underneath a load so that it rests a couple of inches off the ground.

This was done so that ropes could be attached to it for use with a crane, such as when loading a ship, as well as serving to protect the load if it had to scrape across the ground. Dunnage is sturdier than a pallet and is still used for transporting heavy stuff like cut lumber or dressed stone.

EDIT: Okay! I've done a little research to back up my offhand reasoning, because it's been a bit of a slow evening. Morning. Whatever.

The lineage of the forklift is a little difficult to track. Up until about 1910 or so, everything seems to have been winch operated. The earliest design that might be considered a forklift was patented in 1867, and seems to have been a semi-mobile elevator intended for placing items on more maneuverable hand-carts. The next advance came in 1887, with a fixed-axle truck capable of raising its platform about an inch and then moving back and forth. In 1909 somebody created an all steel version which sparked a wave of innovation and patenting, including the 1913 invention of an electrically powered crane on wheels suitable for use by a single workman. The hydraulic load-truck was all tied up in patents and legal troubles until the early 1930's and after that it was anyone's game.

The idea of the standardized, reusable frame pallet (as opposed to the cumbersome and disposable dunnage) seems to have taken off in the mid to late 1930's due to the increasing ubiquity of motorized forklifts. They were adopted by the American military in the Second World War in a frenzy of logistical reorganization prompted by the painfully long supply chains of the Pacific Theatre. After the US Navy and Merchant Marine demonstrated how efficient pallets made things, the rest of the world followed suit.

A skid is by definition, a single faced pallet - which means that it has a top deck, but not a bottom deck. This was quite common in earlier versions which were used for shop floors for work in progress or for goods to be quickly shipped. As inventory accumulation became more important and tiering trucks (earlier forklifts) became prevalent in the 1920s, unit loads increasingly needed to be stacked and so the bottom deck emerged, and subsequently pallets. To be certain, proto-pallets outdated the forklift, but if we hold true the more conventional definitions of the pallet, which includes not only storage but transport, then the case for the forklift first makes perfect sense. Check out my article "What Came First, the Pallet or the Forklift" http://packagingrevolution.net/pallet-or-forklift/