Is there any involvement of brain in reflex action or they are only controlled by spinal cord?

If it is a MonoSynaptic reflex (e.g. Knee jerk", it is taken care of by the neurons in the spinal chord. . The signal will still travel to the brain, but by the time it arrives there action will already have taken place by the Reflex Arc in the Spinal Chord.

Other reflexes ARE initiated by the Autonomic ( more specifically, the Parasympathetic ) Nervous system. Think of sneezing, coughing, urge to urinate, vomiting etc.

These reflexes DO involve the brain, but you can't control them (hence the name Autonomic )

Have a look at: https://socratic.org/questions/which-part-of-the-body-develops-a-response-in-a-reflex-action#541637