Hi,
Nope, I'm saying that Ohms law is really hard to break and even trying to will result in immediate punishment

If the two LEDs are the same type, preferably from the same production batch, they might have a close enough voltage drop over the range of currents a sensible person will subject them to and in that case, you might get away with parallel connection, at least for a part of their life expectancy.
The separate resistor for each led approach is the safest way to go however.
Say you have two blue LEDs and wanna run them in parallel at 12V:
U_drop_LED = ~3.6V
U_drop_Resistor = (12-3.6) = 8.4V
If I_LED_total=40mA
R = 8.4/0.04 = 210 => use 220 Ohm
Actual current will thus be: 8.4/220 = 38.2mA (i.e. 19mA for each)
The power rating of the resistor needs to be: P=U*I [Watts Law] = 8.4*0.04 = 321mW use 1/3W or higher rating.
In the above theoretic case, it would be better to connect both LEDs and the resistor in series, making U_drop_Resistor = (12-2*3.6) = 4.8V
Resistor will then have to be: 4.8/0.02 = 240 Ohm. It will be more likely for you to find a 220 Ohm resistor (from the E12-Row) so making the current 21.8mA (going through both LEDs).
Power dissipated in the resistor is then: 4.8x0.22 = 106mW, so no problem there, not even with 1/8W resistors.
It saves power, it saves LEDs and it saves worries, but you shouldn't use a to low V_drop_Resistor without calculating the consequence if used eg. in a car, where the voltage varies between 12.6V and up to 14.8V under normal operating conditions with a sound battery.
In the latter example, it would mean that R_drop_resistor varies between 5.4V and 7.6V and the current will vary with it, if you use a simple resistor to control the current.