When a City or School gives a group (Club/League) a permit to use the fields, included in that permit is a bit of exclusivity to use the fields. Most permits state that the entire grass area is licensed to the Club/League, not just the area immediately surrounding the lined field and grant the club/league holding the permit the right to dictate who may enter the field. The guy with the dog playing Frisbee in the corner, the family holding a birthday party that spills over from the BBQ/Pavilion to the grass, the Dad and kid playing catch, can all be removed by any agent of the Club/League for just standing on the permitted area.
If this was an SCDSL game, then the following are the policies and enforcement procedure. From the 2017 Game Day Procedure Manual, which forms the basis of the rules and regulations all parents and their invitees (grandpa, Uncle Bob, etc.) must abide by:
Parents/spectators all sit on one side of the field. Home team coach will chose which side of mid-field the home team sits on. Each team parents/spectators will mirror their respective team/coach on the opposite side of the field. Parents are to stay on their side of midfield. PARENTS/SPECTATORS DO NOT SWITCH SIDES AT HALF-TIME. They are to stay where they are the entire game. Please make sure ALL spectators understand what is expected of them on the sidelines during SCDSL games. Clubs, DOC's and coaches will be held accountable for the sideline behavior of their spectators. I would recommend printing a copy of the SCDSL rules in case the referees do not know them. This is a learning process for everyone and the more prepared Team Managers are and the more information they have available to answer questions, the better things will go.
The above is initially enforced by the Team Manager, then the Coach, then the Club/DOC. The procedure manual continues:
Managers should introduce themselves to each other prior to the game. If there are issues with parents/spectators from the opposing teams, parents/spectators are NOT to get confrontational with the opposing team. Managers should communicate with each other and each manager should deal with their own sideline issue. Parents/spectators are NOT to talk, criticize or bother the Assistant Referee on their side of the field. Referee Associations have been instructed that if this does occur, the AR will inform the center ref and the person causing the problem will be sent away from the field.
The Referee is additionally empowered under the SCDSL rules to enforce all SCDSL policies.
Presidio and CSL have adopted a policy of parents sitting on the same side as their team, so I don't know if a similar rule exists for those clubs. What I can also say is US DA, USYS, Cal South all subscribe to the concept of parents on one side and players on the other side.