If you read the Bible, you would discover that it has over 30 references to dinosaurs within it.
You will not find the word "dinosaur" in the Bible because it was first coined in the 1840's. As the Bible is much older then that, and the English versions date from the 1600s, they use the older word "dragon" to identify what we call dinosaurs today. (As did science until the 1840s).
Two of the most amazing are in the book of Job, where God in chapters 40 and 41 speaks of two beast called the Behemoth and the Leviathan. The first matches what you would think of as a "bird hipped" dinosaur (such as a T-Rex or Raptor), while the another is a "lizard hipped" dinosaurs (the four legged kind). These are the two major categories used by science today to classify dinosaurs. Somehow the author of Job knew that centuries before modern science discovered it.
The author also credit to God a very accurate and detailed description of the creatures, including the bird-hipped dinosaurs powerful tail and their massive size. For them to have known all that, either they had seen living dinosaurs, were better scientist then anyone until the 19th century, or they were told of this by God (like the poem claims). Feel free to pick the explanation that makes the most sense to you.
The Bible also speaks of the dinosaurs being destroyed at a time "in the past" by a great disaster that God brought onto the earth. Again the idea that the extinction of dinosaurs is due to a disaster is a modern idea.
Whether that disaster was the Great Flood, or an event that happened prior to the creation of the plant, animals and humans of Genesis 1 is a debated issue by Bible scholars. There are several statements in Genesis (such as God's command to the animals and humans to REplenish the earth) that imply an earlier creation.
The Bible also speaks of "dragon" remains being found in desolate and wilderness places. It uses them as a metaphor several times for something that was completely destroyed by God. Again, the knowledge that the fossils and bones were part of dinosaurs is an idea that science did not discover until the 1800s.
Now science may have added a new piece to the puzzle to support Genesis 1. The current theory is that whatever dinosaurs survived the disaster (flood, meteor, God's wrath, or whatever) have become today's bird. This would place them before land animals and humans on the developmental tree.
Note that Genesis 1 states that first "fish" and then birds where created. Then it becomes a new day (or era) and the mammals and humans appear. Just what science is now suggesting happened. Birds are the remains of an earlier era. Something it appears the author of Genesis knew, and science is just now starting to guess. The only form that the author of Genesis (or Adam and Eve) would have seen dinosaurs in would be in the form of ... birds!! So if they wanted to write about dinosaurs being created before mammals and humans, it is only logical to say the birds were created first.
So the Bible appears to explain dinosaurs (now represented by their only living members as birds) as being created in an era before humans. They were once mighty and powerful creature divided into two major classes (bird hip and lizard hip). They were destroyed by a disaster that God brought on the earth. And are now only found in the form of fossils and bones.
That explanation appears to match well with what science tells us about dinosaurs. So I find no reason to reject belief in a first Man and Woman because dinosaurs existed since the Bible talks so much about their existence.