You certainly want some sort of ground system connected to the other terminal of your
feed system, otherwise it is like only plugging one side of the AC cord into an outlet -
it doesn't work very well.
But running it closely parallel to the antenna itself is not
a good approach, as the
radiation from the two wires tends to cancel. If you made the wire the same length as
the antenna you'd just have a long open transmission line that wouldn't radiate much
Better to run one (or more) wires off in other directions.
... connected to the ground on the balun...
"Ground on the balun"
How are you trying to feed it?
You certainly don't want a balun since it isn't a balanced antenna.