I am reading the book "Robot Control: The task function approach" by Samson et al. (1991), and they're going through some of the math of why $SO(3)$ is difficult to parametrize. They say (paraphrased for clarity):
Consider the open half-sphere $S_+^3 = \{(\lambda_0,\lambda_1,\lambda_2,\lambda_3)\in S^3 | \lambda_0 > 0\}$. This can be mapped to the open subset of rotations with angle not equal to $\pi$. Denoting by $Ret$ the subset of rotations with angle $\pi$, the mapping is a bijective local diffeomorphism from $S^3_+$ onto $\{SO(3)-Ret\}$, it is a diffeomorphism.
Where the set $S^3$ is the set of unitary quaternions: $S^3 = \{(\lambda_0, \lambda_1, \lambda_2, \lambda_3)\in\mathbb{R}^4 \text{ }|\text{ }\sqrt{\lambda_0^2+\lambda_1^2+\lambda_2^2+ \lambda_3^2}=1\}$.
This is fine, but then they say:
We do not yet have a local parametrization because $S^3_+$ is not an open subset of $\mathbb{R}^3$.
Which I can't quite understand. If $\phi: S_+^3 \rightarrow \{SO(3)-Ret\}$ is a diffeomorphism, isn't it a parametrization? Are there other requirements than just being a bijective local diffeomorphism for something to be a parametrization?