This is what the Grant County Health Department has to say about obtaining birth certificates:
"If the birth or death occurred in Grant County you will need to come in to the office or send a written request through the mail. To send a request for birth certificate through the mail you will need to provide the following information; Full birth name, date of birth, full name of father, and full maiden name of mother. You will also need to provide a photocopy of your driver's license or state issued identification card, a stamped, self-addressed return envelope, and a money order for $10.00 for a 5X7 paper certificate and/or $15.00 for a laminated wallet size birth certificate. Please note that we do not accept personal checks."
Kind of defeats the purpose when they want the parents' names and you are trying to get the certificate so you can find them. The state did not start issuing birth certificates until 1907. 1874 is very early, and there probably was no certificate, or any record. Counties weren't really required to keep records until after the 1880s.
Do you know when Samuel died? The Marion library does show an obituary for a Samuel E. Moore in 1952. I did see from the census that he moved around, so this is probably not him, although you never know.