I can't understand how the park is mapped so I don't know how to add these details myself.

Here are some details for both sides:

According to the information board, I think the name is "Conservatoire et Jardin botaniques de la Ville de Genève", and the website is https://www.cjbg.ch

Opening hours (of the park/garden itself):
 * 01 April - 24 October: Mo-Su 08:00-19:30
* 25 October - 31 March: Mo-Su 08:00-17:00