Does wiring 18650 batteries in parallel and/or series increase the discharge rate of the whole pack?
All strings MUST* have the same number of cells.
All cells MUST* be identical (model, date of manufacture, usage)  

Recycled cells may be allowed to violate some aspects of MUST after careful capacity testing and with all cells at the same state of charge at time of pack manufacture. Mixing cells of different models and remaining capacity will often enough attract Murphy's attention. This is not usually wise. 

*MUST === That thy days may be long on the face of the land and that thou shalt not experience the magic smoke and vent with flame. 

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N Cells in a series string:


N cells in series in a string increase voltage to Vstring = N x Vcell 
Current calculations in a string are the same as for a single cell.
eg I_cell_max = I_string_max etc
Ah of N cells in a string = Cell_Ah ** 
Wh of N cells in a string = N x cell Wh
Wh is equivalent to energy so increases with number of cells


K Parallel strings in a battery


Battery V = string V  
Battery currents = K x string current.
eg I_battery_max = K x I_string_max
Ah in a battery - Ah_Battery = Ah_string x K  
Wh in a battery - Whbattery = Wh_string x K


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Back derivable from the above:

Battery of Q cells,
N cells per string  (Ncps)
K Strings (Ks)
Q = N x K
Vx = some Vparameter (Vmax, Vmin, Vmean, ...)
Ix = some I parameter


Ah_battery = Ah_cell x Kstrings   
Wh_battery = Wh_cell x Qcells  
Vx_battery = Vcell x N_cps
Ix_battery = Ix_cell x Kstrings




SPECIFIC QUESTIONS:


  If I build a pack of 6 18650 batteries in parallel which are 3500 mAh each, that would make the pack 21 Ah in total. 


Yes.


  Then I take those 6 and make 13 more packs and wire those in series so I have a total of 78 batteries making 48v (3.7v * 13). This would make a 48v 21Ah battery.


Yes.
But, you meant "12 more" and so "13 total". 


  If each battery has a maximum discharge of 10A. Does that mean the pack has a higher discharge rate? Or does the maximum discharge of the pack stay at 10A?


See above. The discharge rate PER SERIES STRING is the same as per cell, but increases with the number of strings. The fact that the cells are cross connected physically makes the string count a little less obvious, but you functionally have 6 strings, so total allowed current = 10A x 6 = 60A. 



E&amp;OE



** Edited - prior statement should have related to Wh.