Parallel was referring to the how the actual wires are routed to the heater filaments on the tube sockets. In most amp's you see them those wires twisted like the red & black wires here:
View attachment 66402
But certain amps like Soldano's they will run those wires parallel to each other through the sockets like:
View attachment 66401.
The 6.3v center tap being connected to HTR pad on Burger's board has nothing to do with parallel - that just DC elevates the 6.3v AC supply. You could also twist the wires and still connect that center tap to HTR. That said this was my first amp build so burger might chime in and correct me.
Check out
http://www.valvewizard.co.uk/heater.html for some more detail on this.
Other resources that I use a lot during my first build was:
Burgers documentation
Rest of the valvewizard site
https://robrobinette.com/Amp_Stuff.htmUncle Doug's YouTube channel -
https://www.youtube.com/channel/UCuR4hQTXkG_KxozLxwPzEjQ