in the book of Jonah we see God reprimand the prophet for wishing destruction upon the city of Ninevah
God had sent him to warn the city to repent, and Jonah runs off in the other direction by sea, only to have a storm terrify his shipmates and ask whose God had been offended - Jonah speaking up to say 'me, it's my God doing this, thrown me into the ocean' whereupon he was swallowed by a great fish (and likely died) for 3 days, before being vomited up on a shore and making his way to Ninevah
now here's the thing, Jonah was a bit of a big old racis and he really hated Ninevah - but he's possibly just been bleached white by some fish digestive juices, had to make an arduous journey from where he was puked onto land, walks through the city gates looking like he's been mauled by a megalodon, and immediately tells them of their doom… and a figure like that could be somewhat compelling just for his appearance, so the whole city repents in sackcloth and ashes
well Jonah gets mad and goes off in a huff and sits on a hill overlooking the city, and squabbles with God, saying; "i know You are merciful, and i knew that if they repented you would hold back your wrath, and now you're being merciful to them just like i knew you would be'' and he was maaaaaaaaaaaaaaaaaad
so God causes a little vine plant to grow up next to Jonah, and it gives Jonah shade for the day, but the next day it is all withered and Jonah is fuming under the hot sun and bemoaning the fate of the little vine… feeling sorry for it
and God says to him; "Jonah, do you feel sorry for the little plant? Well here before you is a great city of thousands of souls, many of them children who don't know their right hand from their left… and there's innocent animals in there too."
and so God is basically saying that even a wicked city full of wicked people whom a holy prophet could legitimately despise for their wickedness, was still under God's watchful gaze and able to receive God's mercy should He so choose
… but you're probably wondering what my point is, huh?
well go back up there a little bit, and look at the line talking about children not knowing their left from their right - and if you remember in Genesis when early man didn't know the difference between good and evil, he was innocent… and God is similarly saying such babby are 'innocent'
now 'innocent' is a legal definition, and conversely, once a person comes to an age of understanding yet still does what is wrong, they are 'guilty' - so while children may have inherited a Sin nature from our first parents in Genesis, they are regarded by God as 'innocent' before His Judgement Seat
but look to the animals next… God declares animals to be 'innocent', and even mentions their lives as being sacred to Himself when it comes to destroying a wicked city or not
we might also remember that when God demanded sacrifices of the Jews, He required animal sacrifice - not primarily because it cost them to raise an animal - but because only a innocent sacrifice could propitiate God's Wrath, and man is not innocent
well one man was, the God-man, Christ Jesus
yet all that to say, animals are declared legally 'innocent' before the throne of God - being part of the creation which God declared 'good' upon making it, and only suffering as all creation suffers under a curse for the Sin of man
and we also know from the Hebrew words for animals, that from the earliest time they were considered to have spirit or soul - one evidence for the belief that animals have being, being the curse upon serpents for allowing satan to enter them
so if animals are legally declared innocent by God, and if animals have soul, then there is quite a lot of evidence that such beings would continue into the next life - and allowed entry into paradise
but y'know what, if they don't have continued existence after this life, Heaven will be no less sweet for the Believer - since Christ the Creator and fulfilment of Man's soul is everpresent in Eternity, and He is all that a one needs, for He is the source of every good thing
of course, if you are not a Believer in Christ, then you should be more concerned about your own eternal fate rather than that of pets which have passed - for to be eternally separated from He who is the source of all Life, is eternal and tormenting Death…. away from every good thing
TL;DR if this story seems too long;didn't read, you aren't taking this question seriously enough