CSS global :nth-of-type selector (:nth-of-class)

Answer 1

I couldn't get it down to single selector, but with two...

.parent > .element:nth-of-type(2n){
  background:#afa !important;
}
.parent > div:nth-of-type(2n) > .element {
  background:#afa !important;
}

View it on JSFiddle

Answer 2

While it uses JavaScript, you could style the odd element without altering your markup:

$('.parent .element:odd').css('font-weight', 'bold');

​ http://jsfiddle.net/ansonhoyt/MfxYF/