Does the father's current circumstances, such as old age and inability to work or engage in trade, permit him to deal with an interest-based bank by depositing money in investment certificates and accounts that accrue interest?

It is not permissible for your father to invest his money in an interest-based bank, as interest-based transactions are forbidden except in cases of dire necessity. Your father's lack of proficiency in trade is not an excuse for engaging in usurious dealings, as there are Islamic banks available as a permissible alternative. The Islamic Fiqh Academy has affirmed the prohibition of dealing with interest-based banks for anyone who can deal with an Islamic bank.