Is It True that the New Testament Doesn't Mention Infant Baptism?

"For I do not want you to be unaware, brothers, that our fathers were all under the cloud, and all passed through the sea,  and all were baptized into Moses in the cloud and in the sea,  and all ate the same spiritual food,  and all drank the same spiritual drink. For they drank from the spiritual Rock that followed them, and the Rock was Christ.<sup>"
- I Cor. 10:1-4</sup>

I have heard it repeated by Baptists, over and over, ad nauseum, that the New Testament never mentions the baptism of infants, as if that would settle the question. But, as often as it is stated, is it true? No, it's not!
 
Look at the passage quoted above, I Corinthians 10:1-4, referring to the people of Israel during the Exodus from Egypt. This is a familiar story. The Israelites had been given permission by Pharaoh to leave their bondage in Goshen. However, Pharaoh had had a change of heart, and set out in pursuit, intending to retrieve his unpaid workforce. When the Israelites approached the Red Sea (Hebrew, "Sea of Reeds"), their God, Jehovah, made a way for them, dividing the sea to the right and to the left. After they passed through, the Egyptians attempted to follow, but God sent the sea back, crushing the Egyptian troops between two walls of water. The account can be found in Exodus, chapter 14.

Paul uses this experience as a metaphor for baptism. Who was baptized? Adults? Believers? That's what Baptists would have us believe. But what is the actuality? Everyone among the covenant people, men, women, elderly, and children! And this is exactly the case that Presbyterians make. We do not advocate the baptism of those foreign to the covenant, the equivalent of the Egyptians. That is the red herring argument of baptists. Rather, we advocate the baptism of those with a relationship to the covenant, i. e., adult believers, if they haven't been previously baptized, as well as the children within their household. That is, the same that Paul described to the Corinthian Christians.

So, what is the answer to the question in my title? Is it true that the New Testament never mentions the baptism of infants (or children)? Obviously, the answer is no. And, though it isn't the subject addressed here, this text also undercuts the Baptist insistence that baptism must be by immersion. Who got immersed in the Exodus? The Egyptians!

It must be noted that the case I make here only applies to the Presbyterian (or Reformed) view of baptism. It isn't intended to support other views, such as that of the Catholic Church.