Edited: to reflect the correction from the comments.
Question 1: Why are the guts well-defined?
Answer 1: By the JSJ theory there is a unique collection of $I$-bundles (and Seifert fibered spaces) that contain (up to isotopy) all essential product disks and product annuli (and all essential tori). We throw away the $I$-bundles. We further decompose the Seifert fibered pieces to get pared solid tori (which themselves may be thrown away if they are products).
Question 2: Why is the pared guts well defined?
Answer 2: For exactly the same reason.
Question 3: What is the relation between pared guts and the sutured guts?
Answer 3: Pared guts are directed at understanding geometry. Sutured guts have an additional homological condition (the assumption of tautness).
Question 4: Why do we need to consider product disks in the sutured case, but not in the pared case?
Answer 4: We don't need them in either case. Consider the frontier of a regular neighbourhood of the union of a product disk and the sutures it meets.