Will a parallel capacitor placed before a series inductor be capable of self-resonance?
Self-resonance usually refers to a resonance within a single element (L or C) caused by the parasitic inductance/capacitance in combination with the primary capacitance/inductance.

The effect is independent of the position in the circuit, but can vary a bit if the element is in series or parallel orientation due to the impact of the ground plane.



With regard to the resonance of the LC combination as stated in the question, the sequence also doesn't matter.

For example the cut-off frequency: 

\$f_c = \frac{1}{2 \pi \sqrt{LC}} \$ 

is independent of the L/C sequence.