=SUMPRODUCT(MOD(A1:A6,2),A1:A6)

That is for odd numbers... I guess you can use this for even numbers...

=SUMPRODUCT(1-MOD(A1:A6,2),A1:A6)

Test it, more intuitive

=SUMPRODUCT(--(MOD(A1:A6,2)=0),A1:A6)

More intuitive? Well, marginally (at least for me). Since I had no trouble
seeing that Bernd's MOD(A1:A6,2) yields 1 when the processed cell is odd and
0 when it is even, I find no difficulty seeing that subtracting these values
(0 or 1) from 1 reverses the values (they become 1 or 0 respectively) and,
hence, their odd/even-ness (it is nothing more than the principal of
toggling a value between 0 and 1 inside a program where the code line would
be Value=1-Value).

Pity about ISEVEN

I am newly returned to Excel after a very long absence and am puzzled by
this. Why is it that some functions (for example, MOD) can use array ranges
in this way and others (like ISEVEN) can't? Is there some "rule" governing
which function can and cannot?

Rick