Conhecer does mean both ´know´ and ´meet´ in English, and the context is the only way to know which one.
´They can know my mother´ sounds strange because the ´can´ suggests a possible future ability that the women will have. And to know someone, you have to meet them first. It´s always ´meet´ on the first occasion that you see someone new.
But yes, meet should have been suggested as an option.
To be more precise, "podem" can mean "may", not only in the sense of having permission, but also in the sense of indicating possibility (source: https://en.m.wiktionary.org/wiki/poder#Verb_7). Therefore, it seems to me that we can't pin down without context the meaning of "conhecer" in this sentence. "They may know my mother" (meaning, it may be the case that they know my mother) also makes sense as a possible translation. However, it is not accepted.