Excel 2004 appears to calculate correctly, Excel 2011 does not. Problem seems to be that 2011 countif/sumif/averageif do not like having ranges across sheets. When I put the par/score data on the same sheet, averageif works.
try using an array formula, for example:
=IF(ISNUMBER(Score!U3),AVERAGE(IF(Par!C3:U3=3,Score!C3:U3)),"")
instead of hitting return, press command-enter on the mac (ctrl shift enter on PC) to commit the formula; you'll see the curly braces added when you do this.
I did not test extensively, but it seems to work in 2011 and should also work in windows the same. you should be able to adapt to your other columns to suit. good luck!