Are there any months of the year during which the parks are not open?

It is said that the park is not closed for the whole year. From Shenzhen, you may need to take Canton first.
If you take G-trains, you will reach Canton South Railway Station, then you will find free shuttle bus to reach the park.
If you take D-trains, you will reach Canton East Railway Station, then you can take subway line 3 to reach Hanxi Changlong Station. Then you can take free shuttle bus to reach any park.
There are hotels in the park, so you may live in them.