Parallax
A parallax effect is nifty, it looks cool, but does little for usability and efficiency. Some even say it has a negative effect on the User Experience.
My personal opinion is that the parallax effect is good for story telling. E.g. If you're a webdesigner/webdeveloper and you want to explain your work method, you could use a parallax effect. That's because with a parallax effect you normally steer the user in a linear direction (from top to bottom). One page scrollable pages are therefor best suited for this if not the only type of pages suited for it.
So if you have a website that is one page and is suited to steer people linear along a storyline, by all means, use as many parallax effects as you like.
Mobile parallax is a different story. Because a parallax effect kind of hijacks the scrolling, it's not really suited for mobile screen. The screens are to small to fit to many effects in them, some parallax techniques don't work with touch devices and the experience is just not the same. Backgrounds that move at a different speed from the content on top of it is to my opinion, the maximum amount of parallax that you can add to a mobile site.
Also, although I'm not very familiar with it, it seems to me making a parallax website responsive requires quite some work and causes the website to be very heavy, which is not recommendable for mobile websites since loadtime is key.
Flat style
Questions about flat UIs and flat style in general are asked frequently- and also answered frequently on this website and on the internet in general. I could've put several links here to topics that will answer your question, but it's quicker for me to give a short answer.
The flat design is a trend in response to the skeuomorphic design Apple lived by up until iOS7. Why it has better UX? It doesn't necessarily and a lot of people are biased about it. The main apparent advantage is that with flat design any clutter is removed and the users' focus is being directed towards the content. Therefor it is perfect for websites and apps on mobile viewports. The small screens can't handle to much clutter.
The main apparent disadvantage is the affordance. A lot of people are of opinion that flat style buttons don't look like buttons anymore. The buttons don't look clickable anymore.
But all this is mainly based on opinions, both parallax and flat style being good or bad UX. It doesn't really matter what technique or style you use as long as it's used for good reason and suits your projects.