When I read this verse a few days ago, I asked Holy Spirit why He wasn't mentioned, too:
1 John 1:3 NIV
We proclaim to you what we have seen and heard, so that you also may have fellowship with us. And our fellowship is with the Father and with his Son, Jesus Christ.
I think He is teaching me something about the unity of the Holy Trinity. I often think of Yahweh as "Them", but mysteriously, perhaps it is more like "He-Them". And perhaps in some passages, the emphasis is more on the "He" aspect.
I think Holy Spirit showed me yesterday that He is there implicitly in the "fellowship". He makes fellowship possible, and this is probably a heretical thought, but maybe in some ways, He is fellowship.
I think Holy Spirit is also implicitly present in the anointing in 1 John 2:
1 John 2:20, 27 NIV
But you have an anointing from the Holy One, and all of you know the truth. ... As for you, the anointing you received from him remains in you, and you do not need anyone to teach you. But as his anointing teaches you about all things and as that anointing is real, not counterfeit—just as it has taught you, remain in him.
Yahweh, I praise you. You are mysterious and beyond my comprehension. Holy Spirit, I know that you are present everywhere, even when I don't see you. I praise you for your humility. I love you.