There are many covenants which are recorded in the Bible. Some are made by people, some are made by God, and some are made between God and people. The terms Old Testament (Covenant) and New Testament (Covenant) are general terms which are referring to two specific covenants. The Old Testament is a term which refers to the Mosaic Covenant which was between God and the Jews. The New Testament is a term which refers to the New Covenant between Christ and the Church.
The rainbow is not a sign of either of these two covenants. The rainbow is a sign of another covenant which God made, by himself, in front of Noah. That covenant is still in effect today. It is referred to as the Noahic Covenant. The rainbow still appears today because God's promise, to never again destroy the earth by flood, still applies to us today.
Gen 9:12 (NASB)
God said, “This is the sign of the covenant which I am making between Me and you and every living creature that is with you, for all successive generations
The New Covenant did not put an end to all previous covenants. It only put an end to the Mosaic covenant.
Heb 8:6–13 (NASB)
But now He has obtained a more excellent ministry, by as much as He is also the mediator of a better covenant, which has been enacted on better promises. For if that first covenant had been faultless, there would have been no occasion sought for a second. For finding fault with them, He says,
“Behold, days are coming, says the Lord,
When I will effect a new covenant
With the house of Israel and with the house of Judah;
Not like the covenant which I made with their fathers
On the day when I took them by the hand
To lead them out of the land of Egypt;
For they did not continue in My covenant,
And I did not care for them, says the Lord.
“For this is the covenant that I will make with the house of Israel
After those days, says the Lord:
I will put My laws into their minds,
And I will write them on their hearts.
And I will be their God,
And they shall be My people.
“And they shall not teach everyone his fellow citizen,
And everyone his brother, saying, ‘Know the Lord,’
For all will know Me,
From the least to the greatest of them.
“For I will be merciful to their iniquities,
And I will remember their sins no more.”
When He said, “A new covenant,” He has made the first obsolete. But whatever is becoming obsolete and growing old is ready to disappear.
The old covenant being discussed in this passage is the one made between God and Israel and Judah. Israel and Judah were not even born at the time of Noah. In context, the old covenant is the Mosaic Covenant. It's talking about the covenant that He made when he "took them by the hand to lead them out of the land of Egypt." It is talking about The Law, which is another term usually used in the Bible to refer to the Mosaic Law.