In order to earn an S Rank, you must play on Expert difficulty which is unlocked after completing the game on Normal.
Once you've completed the game, you can revisit The Root Pack and play on Expert. This is the easiest boss battle in the game, and I would recommend using the Spread for Shot-B and Super Art I (Energy Beam) as you'll be able to complete the level faster.
In order to earn the S Rank, you must complete the boss on Expert in a certain amount of time (under 2 minutes), with Full Health (3/3), Max Parry (3/3), and Max Super Meter (6/6). On Expert difficulty the boss battles should feel fimiliar although some things are changed. The speed of the battle is increased significantly, so everything becomes extremely fast. Additionally, the boss gains about 25%-50% more health.