You're confusing yourself a bit.
Lets say you have H2C==CH2
In this case you have 3 sp2 orbitals and 1 p orbital per carbon.
In the carbon carbon double bond, the sp2 orbitals form the first bond, and due to their partial s character they overlap well resulting in a high bond strength. The second bond is formed by the p orbital, which has much less overlap resulting in a weaker bond.
The double bond is a combination of both the first single sp2-sp2 bond, as well as the second p-p bond, and so even though there is less energy required to break JUST the p-p bond, breaking a double bond requires breaking BOTH the p-p and the sp2-sp2 bonds. Thus, overall, the double bond requires more energy to break than the single bond, but the p-p bond alone requires less energy to break than the sp2-sp2 bond.
The picture above illustrates the idea behind a hybrid orbital, and shows all the orbitals of interest. The p orbitals overlap side by side and so have little bond strength. The sp(x) orbitals overlap on the fat end, resulting in much more overlap and thus much more bond strength. The double bond contains one sp2-sp2 bond (strong overlap) and one p-p bond (weak overlap).
And finally, to use mathematics to illustrate the point:
sp2-sp2 = 4
p-p = 2
sp2-sp2 + p-p = 6
(sp2-sp2 + p-p) > (sp2-sp2) > (p-p)