Mark 13:32 says:
"No one knows about that day or hour, not even the angels in heaven, nor the Son, but only the Father."
Jesus here is apparently saying the Holy Spirit doesn't know something. How do Trinitarians, who hold the Holy Spirit as a co-equal person of the Godhead, understand this verse?
What the Father knows the Spirit also knows, and the Son in his divine nature also knows.
When our Lord was speaking of the Father he was naturally including "the Spirit of the Father". The Holy Spirit is sometimes called the Spirit of Christ - eg Romans 8:9; sometimes the Spirit of God - eg 1 Corinthians 2:11; and sometimes the Spirit of the Father - eg Matthew 10:20.
1 Corinthians 2:11 says:
For who among men knows the thoughts of a man except the spirit of the man which is in him? Even so the thoughts of God no one knows except the Spirit of God.
What the Father knows the Spirit of the Father knows which is in him. Our Lord was not excluding the Holy Spirit when he said only the Father knows.
One Trinitarian answer is explained in the Church Grammar episode Tough Christology Questions minute 25:28-31:47 on the relationship between Christ's divine and human life.
Your problem:
Jesus here is apparently saying the Holy Spirit doesn't know something. How do Trinitarians, who hold the Holy Spirit as a co-equal person of the Godhead, understand this verse?
Dual nature of Christ (posited by the 3rd council of Constantinople that there are two operations and two wills in Christ) makes it easy: the Holy Spirit (and the Eternal Son) knows, but who from eternity does not inspire the human Christ (a time-bound fleshly "extension" of the eternal Word) to reveal that information to us. Thus in his human nature Christ didn't access that knowledge. With this understanding:
But of that [exact] day or hour no one knows, not even the angels in heaven, nor the Son [in His humanity], but the Father alone.
What is the justification of the above reading? In the Church Grammar episode, Thomas J. White pointed out 2 kinds of knowledge that Jesus has, some of which can NOT be known from his human nature alone (such as knowledge of his own identity), but some which can be known only by prophetic grace when the Holy Spirit reveals the information to Christ's human nature.
BUT the wording of Acts 1:7 (Jesus's answer to the disciples question "Lord, are you restoring the kingdom to Israel at this time?") on the same topic as Mark 13:32 / Matt 24:36, which says "it is not for you to know the times or the seasons ...", implies that Jesus knows some things that he was NOT sent to reveal. So in Acts 1:7 we can deduce that the Holy Spirit didn't impress the human Jesus with this prophetic knowledge since it is not needed for our salvation.
Fr. Thomas White pointed out the pedigree of this reading from the 6th century:
One classical way of reading this goes back to Pope Gregory the Great who was also a theologian of some profundity in the sixth century [which] is to say that the things [Jesus] claims not to know he reveals elsewhere as things he has not been sent to reveal.
CONCLUSION: The Holy Spirit knows, and the Father and the Son who is consubstantial with the Holy Spirit must know also. But in his human nature, Jesus either doesn't know or doesn't want to reveal this knowledge.
OP: How do Trinitarians - who hold the Holy Spirit is a co-equal person of the Godhead - understand this verse?
Since Christ references God the Spirit, but not God the Father, earlier in the discourse at Mark 13:11 as knowing what to say, it (the OP) is a non-starter of a question, unless one wants to say there is only one God the Spirit by cherry-picking verses out of context and ignoring the whole of Scripture.
But when they shall lead you, and deliver you up, take no thought beforehand what ye shall speak, neither do ye premeditate: but whatsoever shall be given you in that hour, that speak ye: for it is not ye that speak, but the Holy Ghost. Mark 13:11
It is an interesting text. Let's break it down to see if it excludes the Holy Spirit.
But of that day and that hour knoweth no man,
Here, man is excluded, the Holy Spirit is not a man.
no, not the angels which are in heaven,
Here the angels which are in heaven are excluded, but the Holy Spirit is not an angel.
neither the Son,
Here Jesus excludes Himself, but the Holy Spirit is not the Son either according to those that believe in the Trinity.
but the Father.
Jesus asserts that the Father knows, but doesn't say He is the ONLY One that knows. Therefore, the Holy Spirit wasn't in the exclusion list.
EDIT: I am editing this as you updated your answer to include a version that says "no one knows". It is important to look at the context. The context is speaking about watching or being ready because MAN doesn't know when Jesus will come. This is important because it is man that must be ready, as their salvation is what is at stake. Jesus adds that the angels and even He Himself don't know when that'll happen, but once again the focus here is the readiness of man. According to the context, and the translation from the King James Version, "no one" refers to "no man" as it's translated in the King James Version.
It would seem to me that the answer to this hinges on which side you stand regarding the Filioque.
Without Filioque. If the Holy Spirit proceeds only from the Father, then it is possible that Jesus could be in suspense about whether the Holy Spirit was yet privy to this information. The sequence of information transfer would be Father -> Spirit -> Son. Thus the Father may or may not yet have communicated the knowledge to the Holy Spirit yet if the Son does not yet know it. In that case we have two choices:
With Filioque. The Spirit proceeds from both Father and Son. The Holy Spirit has no knowledge that the Son does not have. In this case, if the Son does not know, he can be certain that the Spirit also does not know.
This is all complicated by theories of time. Jesus as Son of Man is bound by time and communication occurs over time. All members of the Trinity as divine live in eternity and many think all is simultaneous, so communication and causation are weird. The interface of the Holy Spirit to humanity may also be temporal. In that case, it might be like current distributed systems where information is kept in remote storage. Jesus and/or the Holy Spirit have the knowledge, but it is stored in Heaven, not accessible yet to their projections onto physical reality until needed.
Computer chips have several levels of cache of limited size. When additional data is needed, it is loaded from memory. Jesus as Son of Man has a physical body with an information limit that is not infinite. When he needs to reason about things related to knowledge not present in that human cache, the Holy Spirit can fetch it for him.
As I understand it (I'm a trinitarian but no expert; although it seems clear in scripture that God is three in One the details seem hard to grasp at best):
God is 3 persons with 1 nature, as such it is not necessarily problematic for one of the three persons to have one or more characteristics not shared by the other two. The most obvious example would be, God the Father and the Holy Spirit never had human bodies.
The question then becomes, could omniscience be a characteristic possessed only by God the Father (the person) as opposed to God, the shared nature. It doesn't seem obviously heretical to me to think so.
