Quote from the Park's website: "The park's paths are perfect for walking/cycling" tinyurl.com/Haslam-Park
The park is a good start point, with parking, to explore Haslam Nature Reserve, the Ribble Link, the Lancaster Canal, Cottam's excellent park and cycle paths and new routes via Cottam including Edith Rigby Way.
Despite Komoot's advice that cycling is not permitted on some paths in the park, there's no indication in Haslam Park's official website that cycling is restricted on the park paths in tours 1 to 4 - but always give way to walkers. There's a popular path on the south-east edge of the lake between (f) and (e), not part of the 4 tours, where people stop to view the lake. If you follow that path you are advised to go very slow or dismount - and push the bike up the steps.
It is easier to follow Tour1's satnav route through the park. As explained, the straight line section, (a) to (e), on photo3 doesn't follow a path - it's off-grid. In the western Nature Reserve you can use off-grid paths but are advised to use a route which is visible on the ground. Take care, particularly near walkers and their dogs. In the park do not cycle off-grid. Choose a route using the labelled points on photo3.