The old covenant is not a particular historical period - for example, the period covered by the Old Testament before Christ came. Nor is the new covenant the period after Christ came - or after His resurrection and ascension. The old covenant is the broken covenant between God and man. The old covenant began in the Garden of Eden, when Adam and Eve sinned. It is God's covenant dealings with rebellious, depraved man.
Nobody escapes the covenant.
The new covenant is God's forgiveness of sins on the basis of Christ's shed blood. It is the state of submission to God and obedience to His law. Old covenant religion and new covenant religion run throughout, in the Bible, side by side - and throughout human history. Both the new covenant and the old covenant began in the Garden of Eden. Some tie the old covenant order almost exclusively to Old Testament Israel.