First of all, since you're 16 - you will not invest in anything. You cannot, you're a minor. You cannot enter contracts, and as such - you cannot transact in property.
Your bank accounts are all UGMA accounts. I.e.: your guardian (or someone else who's the trustee on the account) will be the one transacting, not you. You can ask them to do trades, but they don't have to. They must make decisions in your best interest, which trades may not necessarily be.
If however they decide to make trades, or earn interest, or make any other decision that results in gains - these are your gains, and you will be taxed on them.
The way taxes work is that you're taxed on income. You're free to do with it whatever you want, but you're taxed on it. So if you realized gains by selling stocks, and reinvested them - you had income (the gains) which you did with whatever you felt like (reinvested). The taxman doesn't care what you did with the gains, the taxman cares that you had them.
For losses it is a bit more complicated, and while you can deduct losses - there are limitations on how much you can deduct, and some losses cannot be deducted at all when realized (like wash sale losses or passive activity losses).
When you have stock transactions, you will probably need to file a tax return reporting the transactions and your gains/losses on them. You may end up not paying any tax at all, but since the broker is reporting the transactions - you should too, if only to avoid IRS asking why you didn't. This, again, should be done by your guardian, since you personally cannot legally sign documents.
You asked if your gains can affect your parents' taxes. Not exactly - your parents' taxes can affect you.
This is called "Kiddie Tax" (unofficially of course). You may want read about it and take it into account when discussing your investments with your guardian/parents. If kiddie tax provisions apply to you - your parents should probably discuss it with their tax adviser.