Is there any other way of sharing a single email account among two computers?

Yeah, don't most POP3 clients give you the option for "leave mail on server"?

And why don't you just get two email accounts, one for your mom and one for your dad? Most internet service providers include more than one email account as part of the subscription fee. Or, you can do like my wife does, and your mom can just sign up for a Yahoo mail account.