Is this site reliable for replacement parts for x box one controllers?

It depends on the part in question. I've had 4 so far and used 2 (2 1506's and 1 1697 and a 1708 that leaked magic smoke) and they have all broken in different ways; some more serious then others.

My first controller (1537) was made from two broken controllers (Bad RB button but had a good secondary board and the other one had analog drift issues). Eventually this one ended up breaking again and I got rid of it because it's a known issue that you can't fix without replacing the sticks on their own or going through boards that are all likely to fail the same way. I was able to fix the bad board with canned air temporarily but it failed again anyway.

The 1537 is not reliable, and I'd suggest a newer controller (1697 or 1708) if the controller in question is a Model 1537. The main problem with the 1537 that makes it hard to justify repairing is the analog drift problem is the most prevalent on this specific controller.

EDIT (1/13/18): The 1697/1698 (Elite) is better, but even that controller has it's own problems. Outside of established problems like a cheap LT/RT assembly, they are much better then the 1537 controller in terms of durability. It still has analog drift issues, but not at the same rate as the 1537. My 1697 has never had that problem and I've owned mine for ~14 months.

While I have yet to see serious analog drift rates with it, there's a serious design flaw with the LB/RB assembly on the 1697/1698 controllers. The problem is it's a single part and it's so thin that it's fairly common for it to break. The other problem is it doesn't take glue well (I tried on a broken OEM one and it didn't work). If it ever breaks, you need to buy a new one. With that being said it seems to happen most on controllers that were previously abused, but I attribute that to the inherent flimsiness of the part. This is fixed on the 1708 since they beefed up the amount of plastic used, so this is the one I currently recommend if you have the choice of a 1697 or 1708 and you didn't get the controller with a pre-existing problem.

The problem is Microsoft doesn't make the parts available and replaces the whole controller. In order to get parts, we need to buy them from 3rd parties or get used OEM parts. While my aftermarket LB/RB buttons work, it needed to be sanded to fit the controller. The stress broke the cover that keeps it down, so I had to order a new X button cover because of the fit problems I had with my specific part. Thankfully I didn't need a whole shell, but I had to order it from China directly to avoid that. If I wanted a US seller I would have had to buy the shell for one part :(.

In some cases, you're better off with used OEM parts or a donor controller to get what you need. This is a problem with the A/B/Y/X and center buttons since the generic buttons are blank. It's okay to order other buttons like this, but you probably don't want the blank A/B/Y/X buttons. With that being said I'd recommend most parts be purchased from 3rd parties instead of being harvested from a broken controller (Ex: 1697/1708 LB/RB assembly and center cover). Some fitting MAY be required, but it's better then going through multiple failed OEM assemblies.

If you need parts I'd recommend picking up a broken controller or two to harvest the parts you need or buying them from a 3rd party selling compatible replacements.