My dad just retired from Tennessee Wildlife Resources Agency this past year. I don not EVER remember anything about deer not being able to be killed in the early season for meat. Even in my classes at UT talking about wildlife management and hunting, my professors have not mentioned anything about parasites present at early season times.
Things to consider: Early season normally means high temperatures. Keep you kill as cool as possible and get it processed as quickly as you can. Kills left on the back of ATV's or in the bed of trucks for extended periods of time, can lead to the meat turning and causing unpleasant problems if you eat it.
CWD (chronic wastings disease) is a similar to Mad Cow disease but as long as the meat is cooked to a certain internal temperature, there is suppose to be no real risk to humans. However, I personally would do all my research if I hunted in areas prone to CWD. Luckily in TN there haven't been any big confirmed cases
EHD (Epizootic Hemmorhagic Disease) killed off many deer in TN in the past year or two. This disease happens every year to a number of deer but kind of like the common cold for us, deer can get overcome the disease if conditions are right. However, in the past few years it has been extremely dry so conditions have been harsh on the deer. A report came out from TWRA saying the disease should not effect the consumption of the meat. Personally if I shot a deer and, upon closer inspection, found that it looked unhealthy (fur was poor quality, sores on body, etc) I would not consume it. It may not be fatal to do so with EHD or CWD but then again you may spend a little extra time in the bathroom.
Once again, I am not an expert but I have been around wildlife managers all my life and currently enrolled in wildlife program at UT and have never heard any mention of this