"I need money to pay for this cup of tea."
Translation:Necesito dinero para pagar esta taza de té.
'pagar' is 'to pay (for)' and in this case 'para' means 'in order to'. So 'para pagar' is '(in order) to pay for'. The other way around is not grammatical.
If there is a 'to' in sentence that you can exchange with 'in order to' without changing meaning, you have to translate it with 'para'.