Where are you getting your information? par vs pur? Can you provide links to back up your claim that white and better yet RED provides better coral growth?
Well seeing as though a true white light contains all light from the visible spectrum, it's safe to say that no matter what certain wavelength of light is best for coral growth, white light will contain said wavelength, along with all the others in the visible spectrum. That being said, there is no man made LED (that I'm aware of) that can provide a true white light as all wavelengths must be present in order to do so (which is currently not possible). The easiest and most common way to obtain a white light using LEDs is to have a 1:1:1 ratio of red, green and blue LEDs which only provides 3 differing wavelengths rather than the 300 or so needed for a true white light. Even in cheaper LED fixtures that only use those 3 colors to create a white light, you will see more sustainability of photosynthetic organisms while using the white setting than you would if you were simply using only a red, green, or blue setting (given that the fixture emits light that falls within the accepted PUR range of said organism). In higher end LED fixtures you will find more often than not their white lights provide even better growth due to the fact that they use more wavelengths in making their white LED. They do this to try to mimic a true white light as closely as possible.
That alone should be explanation enough as to why a white light provides better growth than any light with 1 given wavelength.
(It's now very late at night and I just spent the last 2 hours researching zooxanthellae)
Regarding red light providing better growth than blue light, here is what I found: (keep in mind, it's very late so this is the quickly jotted simple version, links to nearly all sources will follow)
Zooxanthelae contains 3 kinds of chlorophyll- a, b, and c. Chlorophyll A and B help in photosynthesis by absorbing light at different rates within the visible spectrum.
Although this graph may seem like it gives a clear cut answer, this isn't the case. Chlorophyll B is an accessory to Chlorophyll A meaning it's not as important in the process of photosynthesis, therefore it isn't produced as much (except for in times of little light due to its ability to absorb a broader spectrum of light). Chlorophyll A is more abundant due to the fact that it is the reaction center of the antenna array of core proteins. Although Chlorophyll A is more abundant, chlorophyll B is more absorbant.
Essentially, if you don't have lights strong enough to healthily support your coral's zooxanthellae, they will rely more on their chlorophyll B, while the opposite is true for A. This means that if you use a high output light that adequately meets the needs of your coral's zooxanthellae, a red light will yield better growth than a blue light.
Sources (missing a few)
https://science-edu.larc.nasa.gov/EDDOCS/Wavelengths_for_Colors.html
https://orphek.com/the-correct-light-spectrum/
http://www.differencebetween.net/science/nature/difference-between-chlorophyll-a-and-b/
https://en.m.wikipedia.org/wiki/Chlorophyll_b
It was also delved into somewhat on this thread on a different site.. I didn't get into it much
http://www.thereeftank.com/forums#/topics/220705?page=1