Upload & Sell: Off
Yes, a public park is a public space. Yes, there are ballparks in public parks. Yes, you can go into a public park and take photos to your hearts content. But, the ball parks, if they have been contracted by any organization, is a different story. If a school district has contracted with the parks department to use their sports fields for games, while it is a public park, it is no longer a public event. The school district or the state school athletic association is now in charge of any rules. Why? They pay a fee to use this public park, whether it is for one game, a season, or the whole school year. School districts typically will credential media to cover their events, allowing them on the sidelines. If you are connected to the school or team somehow, Parent with Camera, Booster Shooter, etc, you can most likely get access to shoot at this type of venue. Now, whether you have to buy a ticket or not, since it is a school event or school district event it depends on if they have any rules from photographing from the spectator area. If not, fine, but if they ask for no photography from spectator areas, then you are out of luck. Since you asked for permission, and were asked not to photograph the event, I guarantee you that if you went ahead and started to photograph the event anyway, the administration is within their rights to have you removed and can have you charged with trespassing.
Now, you say you want practice, great, that is help that we can offer. Contact local youth leagues, explain that you are looking to practice sports photography, and would like to come in and photograph their event. If it is a school event, contact the school directly, if it is a youth league, contact their board of directors, if it is a pick up game in the park, ask the players. But once you have been asked to stop, stop, especially in NYC. While there are rules and laws that protect the photographer, the Police tend to over react especially in NYC.
Another option is to contact your state's high school athletic association and apply for a photo credential. They may say no, but you will not know until you ask.
Just because you own a camera does not give you the right to photograph whatever you want. Especially in New York City. In New York, if you want to use a tripod on a city sidewalk, even if it is at 3am in the morning and the street is deserted, you need a permit. And if you want to go far in this business, pissing off the people who can stop you from shooting is not the way you want to go. I have been doing this since 1988 and have watched the rules change many many times. Yes, 15 years ago it would not have been a problem. I could take a pro camera and 300 2.8 lens into an NBA game with no one saying a thing to me, but now, unless you have a credential around your neck, you will find that many arenas will not allow lenses over 3" or no pro level cameras, or cameras with removable lenses.
Not everybody who comes to these boards are going to get access to the sidelines of the sports that they want to shoot....ever, but some will. You have your best chance with youth sports. High Schools can get tougher, especially if you are not media. Team photographer, league photographer, LifeTouch, etc, can make it hard to get on the sidelines. Colleges, can be both easy and hard depending on the school. Some NAIA schools or even D3 schools may offer you an opportunity, but start crossing into D2 or D1 NCAA Schools, and you may find it all but impossible. Pros, probably not, but Minor Leagues, that is a possibility. Club Sports, might work. Just ask to shoot, contact the leagues or teams, school districts, Sports Information Department, Media Relations, etc, and you will eventually find someone who is willing to let you have the access you would like to get.