Electronic – Capacitor in parallel with a resistor between live and full bridge rectifier

It's not clear what exactly you mean by series versus parallel resistor-capacitor filters. Posting a schematic of each would clarify this.

You probably mean series is something the signal passes thru, and parallel is something that works as a shunt. Note that the same thing that is a low pass filter in series is a high pass filter as a shunt, and vice versa.

Basically a capacitor will block low frequencies and short high ones. If you put it in series with a signal then it is a high pass filter. If you put it accross a signal, it will short the high frequencies thereby making a low pass filter. The value of the capacitance and the resistance it is working against tell you the -3dB rolloff frequency of the filter, whether high pass or low pass. This frequency is:

\$ f = \dfrac{1}{2 \cdot \pi \cdot R \cdot C} \$

When R is in Ohms and C in Farads, then f is in Hertz.

The LED driver is self oscillating which gives a simple lowcost approach. In this case it isn't uncommon to not use a chip for low power stuff because orthodox thinking states that chip cost is relatively more significant . Your circuit is a const current source but its not accurate and doesn't have to be . In fact any circuit that relies on inductor characteristics wont be accurate .LED currents do not have to be super accurate but VBE based current sense allows inductor changes without suprises . The small neg temp co of VBE can be beneficial to the thermal management of the LED fitting. I use self oscillating schemes at high power to give switching regimes that give lower switching losses than orthodox current mode but your circuit relies on some inductor saturation to work so is less efficient than normal PWM peak current mode chips anyway.