Hello, Ms.Tara!
Actually, "失くす" and "無くす" are almost same meaning, "lose".
For example, 「私は鍵を無くした(失くした)」 means " I lost my key."
On the other hand, "亡くす" is special word related with death.
For example, 「私は父を亡くした」 means " I lost my father."
In that meaning, we can't say 「私は親を無くした(失くした)」.