Mark 13:32 (as well as Matthew 24:36) records Jesus saying that only the Father knows the hour of his coming. So wouldn't this imply that the holy spirit isn't omniscient if it's a person?
The Bible also says in Revelation that Yeshua has a name known only to him (Revelation 19:12). Does the Father lack omniscience because he doesn't know the true name of Yeshua? I submit that the answer can easily be found other parts of the Bible. Taking two examples:
In the OT, Yeshua wrestles with Jacob and Jacob wins the wrestling match (Genesis 32:25). Does Yeshua therefore lack omnipotence? No, because he cripples Jacob for the rest of his life with a single touch.
Likewise, from the OT, Yeshua struggles against the prince of Persia when attempting to reach Daniel (Daniel 10:12-13). Does Yeshua therefore lack omnipotence? No, because he will defeat sin and death and suffering and the Satan and the armies of the Satan all in one instant simply by speaking (2 Thessalonians 2:8).
Simply put, the Ancient Jewish understanding of omnipotence, omniscience, etc, was that YHWH can choose (and that he has in fact chosen) to restrict himself if he wants to. The Spirit and the Son not knowing the day nor the hour therefore has nothing to do with his status as YHWH. That YHWH doesn't know Yeshua's true name also has nothing to do with his status as YHWH. All three are equally YHWH and all three are equal participants in the humility of the trinity.
(As far as any questions as to why I refer to angel of YHWH in Jacob's wrestling match as being Yeshua, it's because the angel of YHWH calls himself God in Genesis 31: 13. He is also called God multiple times. The angel of YHWH is treated as YHWH yet distinct from YHWH, is commonly mistaken for a man, and Yeshua uses the name given to the angel of YHWH by Hagar as his own name in Revelation 18 (where he calls himself "the Living One"). As far as why I call the man-figure in Daniel Yeshua, this is because the man-figure is referred to as "my lord" by Daniel, thus identifying him as YHWH, as angelic beings are never called lord nor are they treated as being YHWH in any capacity.)
Q1 How do Trinitarians deal with Mark 13:32 in regards to the holy spirit?
I think this is a great question for this board as it allows us to talk hermeneutics. When it comes to biblical interpretation we want to make sure we have read it well from Genesis to Revelation to get the big picture or the entire context. After doing this we will find references to a 3-1 Godhead from Genesis to Revelation that are clear.
Secondly, when it comes to a challenging passage like the verse above we look at it within the context of the entire Bible. If the doctrine of the trinity is clear then and we interpret this verse as evidence that there is no trinity, then we know we are making a mistake in our interpretation.
So, to answer this question specifically, we deal with this passage by using the rest of the Bible (context) to help us view this passage properly.
Q2. Mark 13:32 (as well as Matthew 24:36) records Jesus saying that only the Father knows the hour of his coming. So wouldn't this imply that the holy spirit isn't omniscient if it's a person?
The view of the Trinity is that there is a 3 in 1 Godhead in which there is a unity in deity yet distinctions. God the Father and the Holy Spirit have a divine nature but do not have a human nature. Jesus has a divine and a human nature. They are still 1 God. God having information about a date in the future and choosing to withhold that information from the other members of the trinity does not mean that they are not omniscient. They may know everything (omniscience) except that which God the Father has the ability to conceal from them. With Jesus' ability to know things before they happen and know what people are thinking I wouldn't be quick to write Him or the Holy Spirit off based upon this one verse.