I may be stating the obvious here, but just want to make sure that you are not shooting at too slow of a shutter speed as well. Typical rule of thumb is that the shutter should be around double the frame rate to avoid motion blur. So if you’re shooting at 25p, the shutter should be at 1/50th; if you’re shooting at 50i -that is still 25 fps but broken into 2 interlaced fields per frame, so you would set shutter to 1/50th. Shooting at 50p -set the shutter to 1/100th. Of course this is just a very general rule of thumb but for simplicity, you might want to stick to that. As Laszlo mentioned, shooting at a higher framerate allows you the ability to do smooth slow motion as well. However, (as he also mentioned) it will require more hard drive space …or it will not look as clean (because of trying to squeeze double the picture data into less than double the file size). Also, as I mentioned, the faster the framerate, the faster you should have your shutter, and that means you’ll need more light to shoot, so your video won’t be as clean in low-light situations. If you know you won’t be doing any slow motion, just shoot at 25p.
If you shoot progressive (20p, 30p, 50p, etc.), you don’t need to worry about upper/lower field first, because you are recording/playing back both upper and lower fields at the same time -one full frame at a time. Interlaced video (50i, 60i, etc.) means that each frame is broken up into 2 fields: odd scanlines & even scanlines. HD video is upper field first, SD is lower field first (usually). Interlacing was started because of hardware shortcomings in the early days of video. One could argue that it’s still useful for capturing smoother video, but in my opinion, it’s just an old way of capturing video that doesn’t look as good as progressive because you are never really seeing a whole frame at any one moment in time. Also, if you grab a still from interlaced video, it’ll have horizontal lines through it, or have less clarity (if you use deinterlacing).
Basically, just shoot progressive and don’t worry about deinterlacing, upper/lower fields, etc. Of course, if you already have a lot of footage already shot and it’s interlaced, you will need to treat it accordingly.
As for your motion blur issue, if you have a screen grab of the issue, I might (emphasis on “might”) have a better idea of what is going on. Interlace issues are pretty obvious.
I believe “Parity” typically refers to error-detection. It’s just a way to check to make sure there were no errors in the file you just rendered/recorded/etc. It’s used in a lot of different applications, but usually takes a little more time to verify that all the bits are as they should be before you consider your video render (or capture) successful.
Hope this helps. If you want to learn more about video, there are lots of good educational sources out there. One I remember seeing is https://www.videouniversity.com, though there are probably better sources out there. You could also sign up for a course on udemy.com or linda.com. Get a public library card and linda.com is free. ☺