No, the target phone does NOT need to be rooted for the Android spying software to work.
AndroidMonitor Tracking Application works on both rooted and not-rooted phones and tablets. If you install AndroidMonitor on non-rooted phone and then you root it, AndroidMonitor will continue working normally. Please note that some features, particularly instant messaging monitoring features require the target phone to be rooted. Rooting your phone is easy and it takes just minutes.