Great question! I think it is a matter of style and context.
There is (no) more room on the coach, I'm afraid. This is what I would say. Using 'space' is not wrong, but somehow 'room' sounds more appropriate for talking about 'room' for a person. Others may quite correctly differ.
There isn't nearly enough parking space outside the school. Yes, this is also correct, although many people would say, 'There aren't enough parking spaces outside the school.
I don't think it's a matter of how much room/space is being discussed. I could say,
'There's not enough space in the fridge for this liter of milk.'
'There's not enough room in the whole world for someone as greedy as he is.'
I think they are essentially synonymous as used in your examples.
I hope this is helpful,
PS: Do you know the expression, 'There's not enough room in here to swing a cat!' ?
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