Are the sympathetic and parasympathetic nervous systems exact opposites?

The Autonomic Nervous System is composed of the sympathetic and parasympathetic nervous systems. Each are dominant under different conditions - both are always active but one is dominant.

An important way to understand the the autonomic nervous system is in considering the effects of pharmaceutical drugs.

Sympathomimetic drugs enhance adrenergic function.
Sympatholytic drugs interrupt adrenergic function.

Parasympathomimetic drugs enhance cholinergic effects.
Anticholinergic drugs inhibit cholinergic effects.

For example:

Muscarinic antagonists are parasympatholytic (Anticholinergic) drugs - which reduce the effects of parasympathetic system.

But it is not that simple. They also increase sympathetic tone.

But again it is not always that simple. E.g. for sympathomimetic drugs (enhance adrenergic function) - there is no parasympathetic nervous system effect.

To answer your question, the sympathetic and parasympathetic nervous systems oppose each other sometimes. It depends on which receptors are activated and where.

(E.g.Alpha 1 receptors, Alpha 2 receptors, Beta 1 receptors & Beta 2 receptors).

Source & further reading on pharmaceutical interventions

Source & further reading on receptors