Hey Bastian,
Thanks for posting! I’m not really sure if that feature is part of the theme, it’s quite possible it’s custom coded. Anyway, try the CSS snippet below. It’s kind of an easier way to achieve it.
Basically, how it’s done is simply placing another layer with a background on top of the layer with the background image, pretty much how our background color overlays work. Just that in this case, the color overlay he’s using is in the form of a gradient. If you want to play around with the background gradient, you can search for background gradient generators online and change the background code below.
.fl-row-bg-photo {
position: relative;
}
.fl-row-bg-photo:before {
position: absolute;
top: 0;
left: 0;
width: 100%;
height: 100%;
display: block;
content: '';
z-index: 1;
background: -moz-linear-gradient(top, rgba(0,0,0,0.2) 0%, rgba(0,0,0,0.5) 30%, rgba(0,0,0,0.8) 80%, rgba(0,0,0,0.9) 100%);
background: -webkit-gradient(linear, left top, left bottom, color-stop(0%,rgba(0,0,0,0.2)), color-stop(30%,rgba(0,0,0,0.5)), color-stop(80%,rgba(0,0,0,0.8)), color-stop(100%,rgba(0,0,0,0.9)));
background: -webkit-linear-gradient(top, rgba(0,0,0,0.2) 0%,rgba(0,0,0,0.5) 30%,rgba(0,0,0,0.8) 80%,rgba(0,0,0,0.9) 100%);
background: -o-linear-gradient(top, rgba(0,0,0,0.2) 0%,rgba(0,0,0,0.5) 30%,rgba(0,0,0,0.8) 80%,rgba(0,0,0,0.9) 100%);
background: -ms-linear-gradient(top, rgba(0,0,0,0.2) 0%,rgba(0,0,0,0.5) 30%,rgba(0,0,0,0.8) 80%,rgba(0,0,0,0.9) 100%);
background: linear-gradient(to bottom, rgba(0,0,0,0.2) 0%,rgba(0,0,0,0.5) 30%,rgba(0,0,0,0.8) 80%,rgba(0,0,0,0.9) 100%);
filter: progid:DXImageTransform.Microsoft.gradient( startColorstr='#33000000', endColorstr='#e6000000',GradientType=0 );
}
Let us know how it goes! 
Edit: Sorry, I forgot to mention this only works on rows with background images, not parallax. You can set it to fixed so you get the exact same thing as that site. 
Ben