"I have three classes today."
Translation:Я маю три пари сьогодні.
'три урока' is wrong, 'три уроки' is right. About the other things, like why it wasn't accepted, I could only guess. There's a difference between 'урок' and 'пара', 'урок' lasts for 1 academic hour (45 minutes) and usually used for school lessons, while 'пара' lasts for 2 academic hours and it is mostly used as higher education term. Anyway, report it next time, in casual speech it sounds just fine.
When you say ‘у мене’ instead of ‘я маю’ in Present you should not add copula ‘є’ if you speak about experiencing something like event (‘classes / visit’) or feelings. Use the copula only to stress that you own something. ‘I have a brother — У мене є брат / Я маю брата’ but ‘I have a bad feeling about this — У мене (я маю) погане передчуття щодо цього’ (no copula).
‘Я маю’ is a preferred option anyway.
Excuse me, Spicy Wolf, but you are wrong. I don't know where you are from, but I live in Kyiv, I know both Kyiv and Lviv versions of Ukrainian. It is quite normal to say or to hear "Я маю три пари". Moreover, я маю три ОЛІВЦІ, not олівцЯ, as два, три, чотири require Nominative, not Genitive, unlike the rest of the numbers. Без образ, друже, але ти вліпив бздуру :)