There are many large forests within the Auckland region. Most are plantation pine forests, but there are also large tracts of native bush and forest. Several currently allow horse riding.
Plantation forests are usually pine forests in New Zealand. Some are privately owned and run, while others are Crown Licenced forests. This means that the Crown (we, through the government) own the land and lease out the right to cut the trees to forestry companies. Crown Licence forests and Crown owned forests are most likely to provide public recreation.
Abide by all rules set by the forest manager! Forestry is a dangerous industry, never go into an area where forestry operations are underway.
Native forests are much harder to get access to, due to the conservation arguments. However, even within native forests there are usually traditional routes (kauri, logging or historic roads) that can be opened to horse riders and non-motorised recreation.