Note : I havent found any studies which have done a comparative study between the conversion rate of an non-parallax site vs a parallax site and the points raised below are about how parallax may or may not help in conversion.
The advantage that parallax scrolling offers is the increased level of engagement and the story telling which leads the user along the desired path and higher engagement can lead to conversion. To quote this article from the usablilla blog which highlights the story telling aspect.
Parallax scrolling offers the ideal setting to tell your story in an
engaging and interactive way. Let your visitors take control and let
them walk through your story in their own pace. The different layers
that respond differently to the scrolling behaviour of your visitors
create a sense of depth and even allow for multiple story lines.
Make your visitors curious
The drink Michelberger Booze engages you as a visitor in a very
creative way. At the very top of the site it says: “Just scroll down
gently… When your glass is empty, click to feel the booze.” What is
the booze? When scrolling down the page, you see an artistic still
life setting. When scrolling further, two things happen. The watch
hands rotate and the drink disappears. Then a mask appears which, when
clicking on it, leads you to the booze — a beautiful, animated
illustration that represents the sphere of the drink. Only at the very
end of the site, you get the actual information about the drink.
Still, there is almost no way to get there without engaging with the
Let your visitors have some fun
Another great example of how to engage your visitors is the site of
the KRYSTALRAE fall collection. Different outfits are presented
on one and the same model. The design is very minimalistic, focussing
all attention on the model and the dog in the center of the page. When
scrolling down the page, you can change the outfit yourself, while the
rest of the scene stays the same. This interaction is simple, but fun
and very engaging.
Surprise your visitors
You can also use parallax scrolling to surprise your visitors. For
example, on the Japanese website of the Nissan Note, you get an
entire story about the car, when scrolling down the page. This already
makes it a special experience. However, once you are at the bottom of
the site, there is a link that says: “Try Reverse!” The site
automatically scrolls back to the top, giving you yet another version
of the story.
Similarly parallax scrolling can help focus the user attention to the action trigger or the CTA in a driven approach while keeping him engaged. To quote the above mentioned article :
You can also use parallax scrolling to guide your visitors. You can
guide them into your site, through a story, or towards your call to
action. Tinke, a gadget for fittenss and wellness monitoring, makes
great use of this guiding ability. When scrolling down the page, you
see the device in different settings, explaining different features
and benefits. At the end of the page, it leads the visitor directly to
the call to action of the page: “Shop Now”.
Similarly as mentioned you can drive the users line of sight keeping him engaged longer and hence higher conversion. To quote the above mentioned article
Guide your visitors’ line of sight
You can also guide your visitor’s line of sight with parallax
scrolling effects. For example, when scrolling down on the site of
Guy Vernes, a little character flies in from the left, drawing
attention to the footer information on the right. Without the
animation, this information would otherwise be very minimalistic and
unobtrusive. However, the character leads the visitors’ line of sight
directly to it.
The above advantages are assuming the user is already on the site. That said, parallax sites have SEO problems. To quote this article
How Does Parallax Design Affect SEO? Similar to Flash and AJAX, parallax design presents unique challenges for SEO. Most of these
challenges stem from the fact that all of the content is housed on a
single page, essentially giving you a one-page website.
A single-page website design makes it difficult to optimize the site
for a wide variety of search terms. All of your target keywords must
be concentrated on one page rather than spread out over many, leading
to keyword dilution. Additionally, inbound links can only link to your
site’s single page and not to specific page content.
So the bottom line is that if you are looking for in-page engagement and are not very dependent on SEO, parallax can provide greater engagement and potentially greater conversion as it keeps your users involved for a longer period of time.