Answer
$$r(t)=\lt1, 2+2\cos t,5+2\sin t\gt.$$
Work Step by Step
We know that the sphere of radius 2, centered at $(1,2,5)$, has the equation
$$(x-1)^2+(y-2)^2+(z-5)^2=4.$$
Its projection on the yz-plane is $$(y-2)^2+(z-5)^2=4.$$
So we get the parametrization $$x=1, \quad y=2+2\cos t, \quad z=5+2\sin t.$$
That is $$r(t)=\lt1, 2+2\cos t,5+2\sin t\gt.$$
