The Final Trinitarian Passage
The passage in question is known as the Apostolic Benediction which many commentators (see note 2) state is one of the most explicit New Testament statements of the Trinity. Yet, simply describing three entities does not necessarily indicate deity or equality. Perhaps Paul understands Jesus is only "a" god as some (mis)understand the beginning of the Fourth Gospel.
The opening and ending when considered together demonstrate Paul understands the Lord Jesus Christ is equal to the Father, God, and the Holy Spirit and so God is three equal persons:
2 Grace to you and peace from God our Father and the Lord Jesus Christ. 3 Blessed be the God and Father of our Lord Jesus Christ, the Father of mercies and God of all comfort (2 Corinthians 1) [ESV]
2 χάρις ὑμῖν καὶ εἰρήνη ἀπὸ θεοῦ πατρὸς ἡμῶν καὶ κυρίου Ἰησοῦ Χριστοῦ 3 εὐλογητὸς ὁ θεὸς καὶ πατὴρ τοῦ κυρίου ἡμῶν Ἰησοῦ Χριστοῦ ὁ πατὴρ τῶν οἰκτιρμῶν καὶ θεὸς πάσης παρακλήσεως
The grace of the Lord Jesus Christ and the love of God and the fellowship of the Holy Spirit be with you all. (2 Corinthians 13:14) [Note: verse 13 in some translations]
ἡ χάρις τοῦ κυρίου Ἰησοῦ Χριστοῦ καὶ ἡ ἀγάπη τοῦ θεοῦ καὶ ἡ κοινωνία τοῦ ἁγίου πνεύματος μετὰ πάντων ὑμῶν
The final formulation describes three separate entities (always with the article) each with a separate gift (always with the article) for the Church:

If Paul has the Father in mind in this formulation he is relying on the reader to understand by God he means Father, as in the opening. When the reader recalls and compares the opening, they see the benediction has been constructed to imply Christ's superiority:
- The Lord Jesus Christ is named first and the source of grace
- The Father as such is not included
After opening the letter with explicit reference to grace and peace from the Father, the omission of Father in the closing benediction is unlikely to be an oversight or something Paul has left for the reader to complete. Rather, this is a simple technique to show equality. That is, any argument one is superior to the other is refuted by the passage which shows the other is superior. When considered together, the variation is explained by the equality of Christ and Father.
Moreover, the omission of Father in the ending is consistent with a Trinitarian understanding that "God" is not a singular superior Father but a collective equality of Father, Son, Spirit:
The grace of the Lord Jesus Christ and the love of God [that is Father, Son, Spirit collectively and equally] and the fellowship of the Holy Spirit be with you all.

It is proper to singularly attribute grace specifically to the Lord Jesus Christ (cf. John 1:17) and fellowship with the Holy Spirit (cf. John 15:26). However, since God is love (cf. 1 John 4:8, 16), a Trinitarian would not automatically attribute love exclusively to the Father:

Therefore, within the context of the letter, this Apostolic Benediction has been constructed using grace to convey equality between Christ and the Father from the opening, and love to convey equality between Christ, and God, and the Holy Spirit.
The Introductory Trinitarian Passage
The benediction must be compared to a similar passage in the introduction which some commentators1
see as Paul's anticipation of the ending:
21 and He who is confirming you with us into Christ, and did anoint us, [is] God, 22 who also sealed us, and gave the earnest of the Spirit in our hearts. (2 Corinthians 1 YLT)
21 ὁ δὲ βεβαιῶν ἡμᾶς σὺν ὑμῖν εἰς Χριστὸν καὶ χρίσας ἡμᾶς θεός 22 ὁ καὶ σφραγισάμενος ἡμᾶς καὶ δοὺς τὸν ἀρραβῶνα τοῦ πνεύματος ἐν ταῖς καρδίαις ἡμῶν
This formulation begins as expected, with God first, then Christ, then Spirit. Similar to the ending Father is omitted. Unlike the ending there are four actions which are all attributed to God. The first is in the present participle indicating it is ongoing. The three which follow (with the article) are in aorist participle indicating they are completed and chronologically "out of order" to the first:

The personal experience in being a Christian has been reversed in order to place the current experience of the Church ("you and us") first. This work is attributed to God and Christ. From a Trinitarian's perspective, fellowship of the Holy Spirit is missing. The passage ends with three finished works of God and the Spirit, which from a Trinitarian's perspective is missing Christ.
Therefore, with the exception of the Father, the benediction completes what was missing from the introductory formulation if what is describes is placed in correct order:

Paul leaves the reader to add the love of God to the introductory passage. It was by God's love the believer was anointed and sealed and gave the earnest of the Spirit and it is God's love which is now confirming those in Christ.
Lastly, before the benediction, Paul recalls the opening in a way which recalls the Father:
Finally, brothers, rejoice. Aim for restoration, comfort one another, agree with one another, live in peace; and the God of love and peace will be with you. (2 Corinthians 13:11)
λοιπόν ἀδελφοί χαίρετε καταρτίζεσθε παρακαλεῖσθε τὸ αὐτὸ φρονεῖτε εἰρηνεύετε καὶ ὁ θεὸς τῆς ἀγάπης καὶ εἰρήνης ἔσται μεθ᾽ ὑμῶν
Love and peace are a TSKS construction and following Sharp's Rule, means equality between the two. In this way the reader may understand by "the love God" Paul also means "Father" is included in the benediction since that is how Paul invoked "peace" in the beginning of the letter.
Summary
The oldest confessions of the faith are expressed exclusively in Christological terms:3
it is exclusively belief in Jesus which brings salvation. Most of the Corinthians who became Christians were Gentiles. They were anointed, sealed, and received the Holy Spirit without knowing anything about the Father. For these, the issue is understanding that despite being the exclusive means of salvation, Jesus is not a superior God; He is equally God.
The benediction in 2 Corinthians 13 which begins with Christ is explicitly and implicitly Trinitarian in form. It has ever since been a part of Christian worship tradition.4
Notes:
1. John T. Fitzgerald, The Harper Collins Study Bible, Harper Collins Publishers, 1993, p. 2180
2. Cambridge Bible Commentary, Pulpit Commentary, Barnes Notes, Jamieson-Fausset-Brown Bible Commentary
3. Oscar Cullmann, The Christology of the New Testament, translated by Shirley C. Guthrie and Charles A. M. Hall, The Westminster Press, 1963, pp. 1-2
4. NIV Study Bible, Zondervan, Fully Revised 2002, p. 2414