Can I use only past simple in this sentence?

The sentence, as written, is understandable (although potentially ambiguous - see below). However, if it were me, I would make a couple of changes, to read:

"July 2015: I organised my first family camp in the suburbs of Moscow. At the same time, I closed the family centre that I had opened a year earlier[,] while I was on maternity leave."

Also, I would note that the sentence is ambiguous in that I do not know whether you opened the family centre while you were on maternity leave, or you closed the family centre while you were on maternity leave.

How to re-write it to resolve that ambiguity would depend on which of the two alternatives is correct. The addition of the comma that I have surrounded with square brackets leads to an implication that you closed the centre during the period of your maternity leave, whereas leaving the comma out implies that you opened the centre during your maternity leave, but both are relatively weak implications, and leave the reader somewhat uncertain.

Hope that helps,

Alan.