This achievement is story related and cannot be missed.
Recruiting Officer Parker's help begins with you approaching him in Kid's speakeasy. Start off by saying: "So you've got troubles?". Parker will begin to tell you about his issues, but he will then stop as soon as Trixie begins singing an upbeat song. Exit the conversation and walk up to the sheet of music next to Cue Ball on the stage. Change the song to "Sad Song" and return to Officer Parker. Now select the dialog option: "Tell me your troubles.". Exit the conversation again and head back over to the sheet of music. Select "Secret Song" and go back to Officer Parker. Select the conversation option: "About that secret...", followed by "No", then "What about the people Kid killed?", and finally, "C'mon, pull yourself together." Now leave the speakeasy.
Walk over to Edna who is located in the center of town square and ask her if you can have a copy of her song. Now head back to the speakeasy and equip Edna's Lyrics. Walk in front of Trixie and interact with "I Don't Care" lyrics to swap them out. Walk over to the sheet of music near Cue Ball and select the newly added "You Should Care". Upon hearing this song, Officer Parker will have a change of heart and agree to help you, your trophy will now unlock.