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A few people here have touched on the answer without actually coming out and saying it, so I'll attempt it here.

So, the F Trumpet used in Mahler and Bruckner's day pitch-wise is similar to the Alto Trumpet in F that you see in many russian pieces (Rimsky Korsakov, early Shostakovich, even early Stravinsky), but they were and are completely different instruments supposedly constructed to emphasize different registers (though the alto trumpet did not really help with the lower register much, hence it subsequent disuse). However, the difference that is causing confusion here is that they are notated differently, the old F trumpet transposes UP from its overtone series and the alto trumpet transposes down from its overtone series. So, the way the F trumpet is notated, it's overtone series is an octave lower than we trumpet players are normally used to compared to the fingerings we use, so the pedal C is really not a pedal anymore and can be easily played open, and the G below the staff and the E at the bottom of the staff would also be played open, like on the Horn. In order to write an identically sounding part on both the old F trumpet and the F-alto Trumpet, the alto would have to be notated one octave higher. The Eb trumpet that Strauss used is analagous, hence those "pedal" Cs that he writes. I hope that makes sense.

Jason