It was stated that the bridgemen were a relatively newer strategy by Sadeas. This "war" had been going on for a long time by the time the readers join the war. The strategy is an outgrowth of their observance that the Parshendi don't cross over the chasms to flank or any other tactic once they are engaged.

Building permanent bridges would be an expensive and dangerous task, and I'm not entirely certain how often they'd be damaged by Great Storms. Plus, it was mentioned that the Parshendi had pulled some of the permanent bridges down earlier in the war... not during battle but during their raids during the early parts of the war.

Mobile bridges offer flexibility in the path taken, speed over rolled bridges. Tactically, they seemed to attract the Parshendi's attention away from the regular soldier up to the point of the first charge. Not only are regular soldiers expensive to train, but they're expensive to outfit at well... especially when a lot of that equipment is probably lost in the trenches.

Thirdly, you have to remember that Sadeas very rarely, if ever, is depicted as engaging with the Parshendi. Him and his honor guard are always hanging back. Sadeas unwillingness to place himself in direct danger also provides rear support for the army and the bridges themselves.

Finally, we don't exactly know what the Parshendi are playing at. The assassination. The gem hearts. The possible revelation as to what they really are. There is the fact that the Parshendi actually spring a trap when Dalinar fully commits to what he thinks is the extent of Parshendi tactics. The Parshendi are presenting an image of themselves that seems to be intentional and hiding the fullness of the truth.