Unfortunately, as an AI language model, I am not able to browse the internet or verify the accuracy of information across multiple independent sources. However, I can provide you with some general information about Carthage Park in the state of New York.

Carthage Park is a small community park located in the town of Carthage, Jefferson County, New York. It features a playground, picnic area, walking trails, and a small pond for fishing and non-motorized boating. The park is a popular spot for family outings, picnics, and outdoor activities.

One of the main attractions of Carthage Park is its natural beauty, with scenic views of the nearby Black River and surrounding woodlands. Visitors can enjoy hiking the trails, birdwatching, or simply relaxing in the peaceful surroundings.

In terms of specific points of interest, there are not many notable landmarks or attractions within the park itself. However, the nearby town of Carthage offers several historic sites and cultural attractions, such as the Carthage Free Library, the Carthage Farmer's Market, and the Carthage Area Hospital.

Interesting facts about the area include its rich history, dating back to the early 19th century when the town served as a manufacturing hub for textiles and other industries. The town has also been home to several notable figures, including author Grace Miller White and Olympic athlete John A. Russell.

The best time of year to visit Carthage Park is in the spring or fall, when the weather is mild and the foliage is at its peak. Summers can be hot and humid, while winters can be cold and snowy, so it's best to plan accordingly depending on your preferences.

Overall, Carthage Park is a charming community park that offers visitors a peaceful retreat from the hustle and bustle of everyday life. While it may not be a major tourist destination, it's a great spot for locals and visitors alike to enjoy the natural beauty of upstate New York.