How do the Lilliputians gain favor at court?
As your question implies, a target of Swift's satire in Gulliver's Travels (1726) is the government of Great Britain, particularly the squabbles between the Whigs and Tories, as well as the Catholic and Protestant religions.
When Gulliver is finally trusted enough to be allowed into court, one of the first things he observes is the way in which candidates for high office obtain those offices:
When a great office is vacant, either by death or disgrace . . . five or six of those candidates petition the emperor to entertain his majesty and the court with a dance on the rope.
Essentially, political offices are filled by people who may or may not be "always of noble birth, or liberal education," and they are judged by their ability to dance on a tightrope. The person who jumps the highest without falling becomes the new office-holder.
Swift is, of course, satirizing the tendency of the British court to appoint ministers based not upon their governing skills or intellect but upon their status and social connections. The higher the status of the office, the higher one is expected to jump.
Gulliver also observes another ceremony in which the candidates for office perform a kind of Limbo, with candidates required to "leap over the stick, sometimes creep under it, backward and forward, several times. . . ." Those who succeed at this task are rewarded with silks of different colors, clearly a reference to different levels of honor in aristocratic British society.
Swift's goal throughout each of the four voyages is to point out the flaws not in Lilliput or Brobdingnag but at the British court and within British society. Even though the scenes are very funny, they are meant to correct what Swift views as terrible governance.