Expanding/fixing my comment: Parker-Wolfson's paper came before Kontsevich's compactification with the notion of "stable map". A "stable" ghost bubble necessarily contains at least 3 marked/nodal points, and hence there are finitely many (though need not be bounded above by the fixed number of marked points). With marked/nodal points colliding (not energy concentration), ghost bubbles form that “capture” the points. If there are no marked points, the only compactness phenomenon is energy concentration, and ghost bubbles can form betwixt energy-concentrated components if (for example) such ghosts have three nodal points.
Look 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/component having energy concentration, so in their setup the bubble tree is still finite. In modern language, some of those ghost bubbles are non-stable.