In most cases, you will not need planning permit when installing playground equipment in your own garden. However, there are some factors that may affect this, such as the size and height of the equipment, and whether or not your property is located in a conservation area or has listed building status.
In general, if the playground equipment is a reasonable size and height and does not cause a nuisance to your neighbors, it is unlikely that you will need planning permit. However, it is always a good idea to check with your local planning authority before starting any work.
It is also worth noting that even if you do not require planning permit, you may need to comply with other regulations such as safety standards and building codes, so it is important to do your research and ensure that your playground equipment is safe and meets any applicable requirements.