Nicola, I guess the intended start point is the Main Car Park (blue) next to the Visitor Centre and accessed from the Main Entrance. The location shown on the walksfinder map is closer to the YSP Learning car park (red) which is accessed from Park Lane. There is also a car park at Bretton Country Park Entrance (yellow).
Parking for between 2 hours and all day (any of these car parks) is £8 per car (there is no additional entrance fee).
Number plate recognition is used and so parking can be paid for at any time before leaving or later online.
The entrance to the Main Car Park is here
Nat Grid SE287130
When I visited Poppies: Wave on a Sunday in September all of the car parks near the Visitor Centre (blue, orange and white) were completely full and we had to drive around to the YSP Learning (red) car park (actually nearer to the Poppies) to find a space. Given that this popular exhibit has now been open for some time I would not expect that to be an issue (even on the last day of school holidays).
For a map and further details see Car parking at YSP