If you have one park, you cannot add it to your model. You said you have two management systems, so I interpret your problem as follows:
If you want to incorporate the effect of subjects being in clans in the park, your model would look something like this (taken from your description):
score ~ management.system + (1|clan) for a random intercept
score ~ management.system + (0 + management.system|clan) for a random slope
score ~ management.system + (management.system|clan) for both
The interest lies in the management systems and you didn't mention them being part of some larger population of management systems you wish to generalize the results to, so I see no reason to model it as a random effect.
There is no nested effect here, unless there are multiple measurements of each subject, in which case the model should be:
score ~ management.system + (1|clan/subject)
Or its equivalent for a random slope or both.
I hope this answers your question satisfactory.