Just got the guitar out and checked this - and changed my mind. The "3" in D7/3 refers to the bass 6th string, where you fret the 2nd fret with the thumb. The 3rd note of the scale of D is F# - and the 2nd fret on the 6th string is F#. This would lead naturally into the bass G on the 6th string in the following chord. The music is asking you to emphasise this bass note.

Makes more sense than my previous post, I think.