Is it mostly true that predators or parasites traverse wider spatial areas than their prey or hosts?
The ability of prey to find refuge in predator-free habitats is a fundamental hypothesis for predator-prey stability. The availablility of predator-free patches allows the prey metapopulation to maintain positive population growth in some patches, despite being decimated by predators in other patches. If the predators are far superior dispersers, they would eat all the prey, possibly driving the prey—and themselves—extinct. 

That being said, it's difficult to make generalizations about spatial food webs. In nature, prey, predator, and parasite populations persist due to a multitude of factors, including the spatial configuration of suitable habitats, their dispersal rates, interactions with competitors and mutualists, etc. For example, in a system where carnivores hunt herbivores, sure, the predator may disperse across the landscape, hunting across more than one herbivore metapopulation; in other systems, such as those that may be more spatiotemporally variable (e.g., prone to disturbance and local extinctions), the ability of the prey to disperse faster than the predator could be a crucial aspect of predator-prey stability. 

Some theoretical resources include: Comins et al. 1992, Holt 2002,
Amarasekare 2008.

A foundational experiment is that of Huffaker, who dispersed mites on oranges: Huffaker 1958. Also, take a look at work on host-phage interactions: Schrag and Mittler 1996.

For Google Scholar searches, try various combinations of the following words: spatial, host, parasite, predator, prey, stability, coexistence, food web.