Based on previous comments, I am unclear whether this is a footpath or a shared path (please be considerate to pedestrian and dog walkers if you cycle through, keep your speed down and give plenty of notice 🙂), but this segment is regularly used by Audax events going in both directions, so I'd hope the organisers did their research and did not send us through a footpath... 🙄
The park is extremely well maintained and the road surface very good.
If cycling north-west to south-east be aware of the steep climb at the end, just before the gate: a low granny gear or good legs are thoroughly recommended unless you want to walk up the last part of the lane (conversely, be cautious if you cycle in the other direction as the descent is steep and there is a bend at the bottom).
The gap for bikes and pedestrian on the south entrance is a bit of a squeeze, so be advised you may struggle if you are carrying panniers or similar. On the north side the gap is wider and should not be a problem for most bikes.