Try these formulas...

Odd: =SUMPRODUCT(--(MOD(A1:F1,2)=1))

Even: =SUMPRODUCT(--(MOD(A1:F1,2)=0))

Mike raises a good point about blanks. Here are revisions to the above which
will account for the blanks...

Odd: =SUMPRODUCT((A1:F1<"")*(MOD(A1:F1,2)=1))

Even: =SUMPRODUCT((A1:F1<"")*(MOD(A1:F1,2)=0))

Rick