Sam Gardiner said it would be a "fitting tribute" to the 18 British soldiers who died there in a double bomb attack in 1979.
The bridge, which provides a cross-border link between Co Down and Co Louth, was given the go ahead by Finance Minister Sammy Wilson last week.
Upper Bann MLA Mr Gardiner described it as "unnecessary".
He continued: "The Narrow Water Bridge is, to my mind, unnecessary and is the product of the kind of crazy economics and white elephant projects which Sinn Féin are given to.
This is the Sinn Féin technique
"There is a perfectly good road between Belfast and Dublin a few miles away and the local economy on both sides of the border there does not justify such a costly infrastructural project at this time of cut backs.
"Now that this white elephant bridge is going to be built, I believe it would be a fitting tribute to the memory of the mass murder of 18 members of the Parachute Regiment by the Provisional IRA in 1979 if the bridge were named in honour of the Parachute Regiment.
"It could be the Parachute Regiment Bridge."
The £17m project - which is funded largely from Europe - was approved on 28 May.
The proposed 660 metre single carriageway, cable-stayed opening bridge has been designed to span Carlingford Lough, close to Warrenpoint, Co Down.
It will cross from the Newry River at Narrow Water to Omeath Road at the townland of Cornamucklagh, near Omeath village, Co Louth.
The bridge is expected to deliver construction jobs and an economic boost for the area.
Sinn Féin MLA Caitriona Ruane described Mr Gardiner's call as "bizarre".
She continued: "The intervention by Sam Gardiner calling for the name of the Narrow Water Bridge to be called after the British Paratrooper Regiment is bizarre and divisive.
"The campaign for the bridge has been fought by all sections of the community on a cross border basis and the name of the bridge should reflect that. Sam stated he has not consulted with the local population before making his remarks and I am sure that Newry/Mourne and Louth Council will decide on an appropriate name when it is complete."