Am I really doing this the wrong way? Do they need to be in series instead?
Yes, by having only one constant current driver, you then have no defined control mechanism for equal current flow to each parallel LED, your just assuming somehow that each will take its equal share, and while that may be your wish and desire that doesn't make it so. Kirchhoff has defined how current and voltage works in series and parallel circuits and is worth a review: Kirchhoff's Laws for Current and Voltage
And if so, can anyone explain why they would need to be in series and parallel doesn't work?
Well according to one of Kirchhoff's rules in a series circuit the current flow is equal at all points in the circuit. So if three leds are wired in series and 700ma is the circuit current flow then of course 700ma is flowing into and out of each led, so they all operate at the same current. But of course as the desired LED current is 700ma for your LEDs, your constant current driver needs to run at 700ma output only, you can't use one that outputs a constant 2800ma output.
I don't exactly see why it would make a difference as long as they are being fed their correct voltage (for example say there are 4, 3.5 volt led's, either feed 3.5 volts to them all in parallel, or feed 14v to them in series, why would there be a difference).
As other have said, you need to put LED forward voltage drop specifications into the background of your thinking, you need to keep controlling the current to a desired value in the forefront of your thinking. You don't operate or control a led by manipulating the circuit's voltage supply, you control the current by whatever method you deem to use, be it a constant current driver (which manipulates applied voltage to maintain a desired current) or with a series current limiting resistor (which just sets a max current that can flow given a constant applied voltage) which is not recommended for high power LEDs unless you run them less then their max rated current spec.
And if so, can anyone explain why they would need to be in series and parallel doesn't work? I don't exactly see why it would make a difference as long as they are being fed their correct voltage (for example say there are 4, 3.5 volt led's, either feed 3.5 volts to them all in parallel, or feed 14v to them in series, why would there be a difference).
If you wish to run LEDs in parallel then each LED needs it's own constant current driver rated at the leds operating current, 700ma in your case. If you run the LEDs in series then a single constant current driver will work with only the caveat that the driver must have a maximum output voltage capability of at least the combined forward voltage drop sum of the number of leds in the series string.
Lefty