As long as the transistor you are using can handle the total power across it for the duration you are placing it under load, you can put as many LEDs as you like across it.
I recommend using series-strings of LEDs if you have a higher voltage you're starting with, so that as much power as possible is going into the LEDs and not across the resistor and transistor.
If you choose to use parallel LEDs, use separate resistors for each one, rather than a smaller resistor for all of them, so that you don't have to worry about differences in the LEDs' forward voltages causing the ones with lower voltages to rob the ones with higher voltages of their power.
If you are pulsing LEDs and transistors with VERY short bursts of power at low duty cycles, and don't actually exceed their absolute maximum ratings, you can even leave out any dropping resistors, which ensures more power thru the LEDs and less wasted in-circuit. But if you do this you should try to calculate out and/or measure the voltage drops across the LEDs and transistor(s) so that they take up the full voltage provided to them, with little or none leftover.
You don't have to use FETs instead of regular Bipolar transistors, though since it is an on-off application the FETs will probably be more efficient. If using FETs, the RDSon value and the Gate-turn-on voltage (and charge) are two of the most critical values you need to select them by, as well as the actual maximum D-S voltage and current.
There are some automotive LED driver chips from Maxim and others that use FETs designed for the purpose of driving LEDs for markers and turn signals, as well as interior lighting, and some of them can even do buck/boost on the supply voltage to keep the levels perfect for the LEDs to keep them always at the same brightness (forward current). Most of them are surface-mount, though, so keep that in mind when designing with them.
Remember when using larger capacitors that the inrush current to charge them at first will be significantly higher, so any component in the path of that inrush current needs to be able to handle it.