Elaborating on my comment: This paper of Parker-Wolfson came before Kontsevich's compactification with the notion of "stable map". Then a ghost bubble necessarily contains marked points (in lieu of nodal points), and hence there are finitely many, bounded above by the fixed number of marked points. If there are no marked points, the only compactness phenomenon is energy concentration, hence no ghost bubbles. With marked/nodal points colliding (not energy concentration), ghost bubbles form that “capture” the points.
Looking at Lemma 4.2 (in Parker-Wolfson) and how it is used later on: Their ghost bubbles sit in a chain which must terminate at a bubble having energy concentration, so in their setup the bubble tree is still finite. In modern language, those ghost bubbles are non-stable.