I'm having trouble understanding the questions. I think there may be a vocabulary problem - perhaps a confusion between the words "lift" and "tye."
A yard normally (there are plenty of exceptions) has three pairs of cleats on it. (The "cleat," I'm referring to, in this case, is a shaped chunk of wood that's nailed to the yard - not the T-shaped fitting that rigging lines are secured to.) Two big cleats, usually referred to as "sling cleats," (that's the term James Lees and Brian Lavery use to describe them) are nailed to the front of the yard near its center, the distance between them being equal to 1/4 the diameter of the yard. (Mr. Lees is talking specifically about warships, but it seems safe to assume that a merchant vessel's yards would be similar. Mr. Lavery seems to have made that assumption.) The "yardarm cleats" were nailed to the fore and aft sides of the yard, one pair at each end. (The word "yardarm" is one of the most frequently misused in the nautical idiom. A yardarm, or yard arm, is the outermost portion of a yard. Mr. Lees says "the yard arm during all periods was one twenty-fourth the length of the yard." I think it may have been a bit more complicated than that, but the point is that each yard has two yardarms - one at each end. And the inboard end of the yardarm is defined by the yardarm cleat.)
A "halyard" (or "halliard") is a heavy line that's attached to the center of the yard, and is used to haul the yard up and down the mast. (Strictly speaking, in the case of a large yard the job is usually done by a tackle consisting of two ropes. The one that's actually secured to the yard is called the "tye." It has a block of some sort in its lower end; the "halyard" connects that block with another one on the deck or in the channel.) A "lift" is a line leading from the yardarm to the masthead. (Each yard has two lifts - port and starboard.) The purpose of the lift is to keep the yard horizontal, or, under certain circumstances, to tilt, or cant it. The end of the lift (or, in the case of a large yard, the block through which it leads) is slipped over the yardarm, and is prevented by the yardarm cleat from sliding inboard. If there's a sail above the yard in question (i.e., in the case of the lower yard), the lift block is usually stropped together with that sail's sheet block. (Mr. Lees suggests that, up to about 1660, there were in fact two pairs of cleats at each yardarm - one pair to keep the lift and sheet blocks from sliding inboard, and one pair to keep them from falling off when the yard was canted, as it frequently was in those days. That arrangement is shown in Mr. Lavery's book on the Susan Constant - p. 107, drawing I7. On p. 110 of the same book he shows the lifts; they're numbered 2 and 7 in drawing J7.)
The tye, or halyard, is secured to the middle of the yard - between the sling cleats. In some variations two lines, or both ends of the same line, would be secured there - both inside the sling cleats. They're shown in Mr. Lavery's drawing J7, though that drawing is a little hard to follow. Drawing J1 shows the tye and halyard arrangement pretty clearly - with both ends of the tye secured to the yard. Put J1 and J7 together and you've got it.
The parrel is a simple gadget that holds the yard to the mast - as shown in Mr. Lavery's drawing J6. (To my eye the ribs in that drawing are too thin, but the principle is clear - bearing in mind that the trucks are omitted.) The locations where the parrel ropes are attached to the yard are determined by the diameter of the mast. They have to be set up in such a way that the parrel trucks roll on the mast.
I've whined more than once about the failure of Heller, in its large-scale sailing ship kits, to represent the parrels. (The Heller H.M.S. Victory and Soleil Royal kits make no provision whatever for fastening the yards to the masts. Apparently they're just supposed to sort of hang there.) In fact the parrel is a crucial part of the ship's propulsion machinery. The wind fills the sail, the sail pulls the yard, and the yard uses the parrel to pull the mast. The mast, by way of the standing rigging as well as its own connection to the fabric of the hull, pulls the ship.
Off the top of my head I can only think of one plastic kit that makes even an effort to represent the parrels accurately: the Airfix Wasa. Revell usually used "snap rings" or other simple devices to hold masts to yards. (At least they were fastened together somehow.) It strikes me that molding a parrel, with individual ribs and trucks, wouldn't actually be at all impractical on any reasonably large scale - but to my knowledge no plastic kit manufacturer has tried it. The aforementioned Wasa, as I remember it, has passable representations of parrels molded integrally with the masts.
I guess every modeler has his/her own preference when it comes to the sequence of rigging. My own usual preference is to set up the masts, then the standing rigging, then the yards - but that's certainly not the only way to do it. What is important is to rig the standing rigging - specifically the shrouds, stays, topmast shrouds, and backstays - before installing any parts of the running rigging that depend on the masts being lined up right. If you install much of the running rigging before the shrouds and stays are in place, the running rigging will go slack when the standing rigging is set up.
I'm not sure whether I've answered the questions or not - but I hope all that helps at least a little. Good luck.