As a topological space, this is homotopy equivalent to $\mathbb{S}^2,$ so the homotopy groups are whatever they are for $\mathbb{S}^2.$ As an orbifold, the fundamental group is $\mathbb{PSL}(2, \mathbb{Z}),$ while the higher homotopy groups vanish, since the universal cover is $\mathbb{H}^3.$

As a topological space, this is homotopy equivalent to $\mathbb{D}^3,$ so the homotopy groups are whatever they are for $\mathbb{D}^3.$ As an orbifold, the fundamental group is $\mathbb{PSL}(2, \mathbb{Z}),$ while the higher homotopy groups vanish, since the universal cover is $\mathbb{H}^3.$

NOTE Thanks to HJRW for the correction.