I will concede to the minor point that the only Artillery which may have actually crossed "with Parson Brigade" may have only been that single battery. That the artillery train of 25 guns and their supplies in question could have crossed a few days earlier.

However if you were the commander of such an operation as crossing such a valuable train of ordinance supplies, would you not have wanted to be in close contact to a protective covering force as you possible could? Not only for the protection which a nearby brigade of Infantry might provide you, but also the confusion that multiple reports of these crossing would cause the enemy intelligence. IF those guns were being crossed under those circumstances they certainly would not want to risk their lost if at all possible. Those guns and supplies probably were more important that Parson Brigade of Infantry at that point.

Dodging Union navy patrols of the river and supply convoys going to the Union army and navy at Vicksburg would not have been an easy task for even a small party let alone the size of forces we are talking about.

One of the points that convinces me that this artillery did cross the river is that Gen'l Curtis makes no mention in his dispatch the he believed he was chasing a Brigade of Infantry with artillery. He appears to believe that his forces were chasing a supply train containing this artillery. That if his forces were able to catch up to it they would be able to effect it capture. If his intell was good enough to report the crossing of 5 pieces of artillery with Parsons Brigade, why did they now also report the presence of Parson's Infantry crossing also?

I do not believe that Curtis knew anything about Gen'l Parsons crossing as he makes no mention of it until much later.