WRT the NYC and IRC/Erie/EL being parallel between NT and Lockport, it depends on your definition of parallel. They were both primarily straight lines, however as I said they ended up in majorly different locations within the Lockport area and somewhat different areas of NT. The NYC branch came out from their NT yard (which is essentially the existing Niagara Branch) crossing Oliver Street between Thompson and Christina, and followed a straight-as-an-arrow path (compass North-Northeast) to meet the FRR at Junction Rd. The IRC/Erie (what is the current Lockport Branch) exited the Erie yard at the location of the current branch crossing Sommer Street, crossed the NYC branch south west (compass) of the Twin Cities Memorial crossing, and followed an equally straight path (albeit more compass Northeast) until Hinman Rd in South Lockport (where the Gulf line diverged) and went into the end-of-line at a yard neighboring the canal. Basically the trajectories they separated at when they crossed at Twin Cities was the trajectory they continue on into Lockport, and this divergence can be seen in current maps by looking at the scar left by the NYC branch (the "Wurlitzer branch" as named by Conrail and before CSX removed it completely, as it was truncated back to the Wurlitzer factory under CR) https://goo.gl/maps/K1MDdL6LPtVfkYiv5
WRT the bridge in question, I can only specify names as Conrail referred to them as in their track charts, so apologies if the originating railroads' names for these lines are incorrect.
The bridge was part of the Olcott Branch (per Conrail's name, although why a branch not much longer than the creek is wide needed its own name is beyond me) was a connection between the Erie's Gulf Line I previously mentioned, and the NYC's Lowerton Branch, which was the line that serviced the industries on the north side of Eighteen Mile. The Gulf Line ended at a dead end after crossing Gooding Street, just before the hard bend of Harwood/Jackson streets. This bridge would have been a reverse move off the Gulf at that dead end and crossed over to the Lowerton Branch.
The Lowerton followed Mill and Chapel streets before making a hard turn at the canal, where it followed Van Buren for a ways before crossing the canal at the same location where Lake Ave now crosses (Lake Ave originally continued straight to a diagonal intersection with Van Buren and crossed there) before connecting to the Falls Road. I think the bridge you're asking about would have been Erie owned, with I'd assume Erie having trackage rights to serve the customers on the Lowerton. This was also the ONLY interconnection between the Erie/IRC and the NYC within Lockport, that I can find.
In the Sanborn thread I believe I mistakenly said that Conrail had built the "Olcott branch" as it didn't show up in the earliest available track charts from the '70s, however this is obviously in error as the line shows up on topos from much earlier and has a build date of 1889.