Upon being exonerated of the charge of misconduct, Hart's Battery was reorganized in August 1862. This second organization had little or no connection with the original battery, whose members apparently joined other organizations. The second organization of Hart's Battery is the basis of the roster on the Gerdes page. It was composed mostly of transfers and details from infantry regiments stationed at Camp White Sulphur Springs in Jefferson County. This battery was captured at Arkansas Post, following what most historians agree was courageous and professional conduct by the battery during the battle.
The third organization of Hart's Battery is essentially the second organization following its exchange after Arkansas Post. No comprehensive roster of the third organization has been found, but other records indicate that it was composed primarily of Arkansas Post survivors from the second organization, supplemented with details from other organizations, as well as some conscripts.
Ed, your "quick question" about exchanges and paroles would require a major effort to answer. It's one of the most discussed aspects of the Civil War. One of the most detailed discussions can be found in the Southern Historical Society Papers, which I believe is available on-line.
As to the obligations of paroled and exchanged prisoners, that is pretty straight-forward. Prisoners could be released upon signing a "parole of honor," under which the prisoner was released from custody and permitted to return to his home, to be undisturbed so long as he agreed to obey the laws in effect at the time and refrain from rendering any aid to his military. The Confederacy usually granted the returning parolee a furlough, followed by orders to report to a parole camp to await exchange.
Exchanged prisoners were free to resume their military duties immediately upon reporting back to their units.