I will be attaching a 33' folded dipole for RX (made from 300 ohm
twinlead) to the underside of my wood deck. It will be horizontal.
Does it matter if the parallel conductors are in the horizontal or
vertical plane?

Short answer: No

Think of it this way, what's the separation distance, relative to a
wavelength?
If it's less than 0.05 wavelength, then any difference is almost certainly
unobservable at any significant distance.

You won't be using metal staples though, right?