I want to make a payment to a Sparkasse account in Germany. I am currently in Germany but do not have a bank account. Can I make this cash deposit directly from one of the Sparkasse branches?
Yes, that should be possible (though any given financial institution or branches thereof may chose not to provide this service). According to this source the Sparkasse offers it (edit to add: per OP's comment the Sparkasse offers it only for accounts with the same local branch). It is called Bareinzahlung auf fremdes Konto. The process is done using a Zahlschein. Obviously you will need the bank details (IBAN, BIC) of the benefited.
Note that special scrutinity concerning your identity may be applied to prevent money laundering - be prepared to identify yourself with ID or passport. There will also be additional fees higher than depositing to your own account (edit to add: per OP's comment the Deutsche Bank has a fee of 15 Euros per transaction).