Will there be pets in heaven?
Before answering this question, we need to clarify how Scripture speaks of heaven. The term can mean several things, depending on the context of its use.
In 2 Corinthians 12, the Apostle Paul refers to a man, likely himself, who was caught up to the "third heaven." Expressing ancient Hebrew cosmology, Paul conceived of the heavens in three layers: the first heaven (the atmosphere around us), second heaven (the sun, moon, and stars above us), and third heaven (the abode, or dwelling place, of God beyond us). Your question no doubt refers to this latter sense.
While the Bible speaks of the disembodied spirits of God's children enjoying the presence of God immediately upon death, this is only a temporary and intermediate state (2 Corinthians 5:6-8). One day, at the visible return of Jesus Christ, God will ultimately raise the souls of these believers in new, physical, and glorified bodies (1 Corinthians 15). Their final dwelling place will not be in the third heaven per se, but rather with God on a new earth under a new first and second heaven (Matthew 24:29-31; Revelation 21). The presence of sin banished, there will be no more sorrow, suffering, or death (Revelation 21:4).
Thankfully, there is credible evidence to suggest that God's plan for the new earth involves the presence of animals. First, they were part of God's original and good creation (Genesis 1 and 2). It, therefore, seems possible, perhaps even likely, that the eventual restoration and fulfillment of creation will be comprehensive - extending to all creatures.
Second, and with that in mind, the Apostle Paul wrote creation itself longs for (and will benefit from) the full redemption of humanity. We read these words in Romans 8:19-21:
For the creation waits with eager longing for the revealing of the sons of God. For the creation was subjected to futility, not willingly, but because of him who subjected it, in hope that the creation itself will be set free from its bondage to corruption and obtain the freedom of the glory of the children of God.
In so doing, Paul seems to anticipate a full liberation and reconstitution of creation in the coming fullness of God's Kingdom.
Finally, Paul's expectation builds on the prophetic teaching of the Old Testament. Prophets like Isaiah routinely used animal imagery to describe the coming age. In Isaiah 11:6-9, we read:
6 The wolf shall dwell with the lamb,
and the leopard shall lie down with the young goat,
and the calf and the lion and the fattened calf together;
and a little child shall lead them.
7 The cow and the bear shall graze;
their young shall lie down together;
and the lion shall eat straw like the ox.
8 The nursing child shall play over the hole of the cobra,
and the weaned child shall put his hand on the adder's den.
9 They shall not hurt or destroy
in all my holy mountain;
for the earth shall be full of the knowledge of the Lord
as the waters cover the sea.
We find comparable passages elsewhere.
Does this mean our particular pets will join us in the fullness of God's kingdom? The Bible does not say, and we should be careful to avoid too much speculation. Even so, nothing in God's Word necessarily precludes it.
One thing we do know for sure: there will be no sadness in heaven, only complete and lasting joy (Revelation 21:4). Whatever our Father has decided on this matter, we will be at peace.