Difference between Stockholders' Equity and Stockholders' Equity-Parent?

Most likely, this is referring to the parent company. The rules regulating when the parent company's stock must be reported on the subsidiary's financial statements depend largely on how much of the subsidiary's income is attributable to the parent company. The rules governing consolidated financial statements are quite technical and a bit complicated. Usually, whenever an item in the financial statements says "parent" it is showing the parent company's influence on that item. If the parent company had a substantial influence on the subsidiary's financial statements, then the subsidiary will usually show the part attributable to the parent company on a separate line.