You're right in saying that in English "moreover" is not the same as "on the other hand", of course, but in Spanish, as explained in the lesson, we can use "por otro lado"/"por otra parte" with two different meanings:
1. as a contrast (in English "on the other hand")
Por un lado me da envidia, pero por otro lado, me da pena.
On the one hand I envy her, but on the other hand, I feel sorry for her.
2. as an added idea (in English "moreover, in addition, plus...)
En la excursión de hoy vamos a disfrutar mucho de los paisajes. Por otra parte, visitaremos dos iglesias muy antiguas.
On our trip today we are going to enjoy the landscapes. In addition, we'll visit two very old churches.
This is the translation that we have in the English sentence in this case: "moreover"(not contrasting, but adding an idea)
I hope this clarified it.
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