The Spirit’s role

Philemon 1:3
Grace to you and peace from God our Father and the Lord Jesus Christ.

Why does this salutation not mention the Holy Spirit? Is Paul’s understanding of the Trinity deficient?

Not at all. The Father’s role is external. The Son’s role is external. The greeting is, in one sense, external, coming to Philemon from the outside. The Spirit’s role, by contrast, is internal, dwelling within the heart of Philemon and every believer.