There may be some confusion as to what defines trackage and haulage rights... below, I've paraphrased an earlier posting I made on this in '04:
Railroads cannot arbitrarily have crews "running through" on foreign lines. In order to do so, the two companies need to first establish trackage rights. In this case, PAR would have to grant NS the right to use their power and crews to effect freight movements over their line east from Mech'ville. In exchange for this right, the host railroad (PAR) would be paid a fee, usually in cents per car-mile. NS would assume car-hire costs for that portion. Trackage rights are usually in effect in perpetuity, and can only be cancelled by mutual consent. In other words, once PAR approves it, they're stuck with it unless NS also wants out. Rights can be limited to discrete traffic, such as particular coal movements. Keep in mind the fact that NS would have to establish a crew pool that is qualified on PAR, absorb taxi costs for NS to their home terminals, etc...
An alternative to trackage rights is haulage rights. In that case, the host railroad will move freight that is "in the commercial account" of another railroad for a preset fee (many rates are contracted and set individually based on competitive issues). Say MMA decided to establish haulage rights on GRS between No Me Jct and Mechanicville, and PAR agreed. GRS would move MMA's traffic on GRS trains with GRS power and crews. GRS would charge on a per car basis, say, $1,000, regardless of commodity, origin or destination. The cars would never be taken into their account, that is MMA would be responsible for car-hire and PAR would not be privy to what is in those cars (with the exception of hazardous materials) or how much MMA is making on the line-haul. A car shipped from Millinocket would be routed MMA-mechanicville-CPRS.
There are a host of variations to these rights, all negotiated to maximize operating efficiency and protect the commercial interest of both parties.
That said, I highly doubt PAR has granted either right to NS for coal movements to Bow. They've traditionally made a killing on pre-existing linehaul rates over Michy/Rott Jct interchanges.
What I can say with certainty: The presence of trackage or haulage rights can't be determined by who's locomotives or cars are in the consist.