Is the Lowest Possible Spin of Particle p 2 in Decay Scenarios?

[MarkovMarakov]

Does the following argument correct?

Suppose there is a particle $p$ that can either decay into $ \{$a spin-1 and a spin-0 particle$\}$ or two spin-0 particles, then the lowest possible spin of $p$ is 2. This is because we need the spin to be even and large enough to accommodate the spin-1 product.
(Assuming that the total ang momentum is conserved in the decay.)


Thank you in advance!

 

[MarkovMarakov]

ADDED:

$p$ is such that $p\to \pi^-+\rho^+$ and $p\to \xi+\xi$ where $\xi$ is a particle of spin 0. I want to know the lowest possible spin for $p$.
I am assuming that conservation of total angular momentum and parity holds.

By the way, would the intrinsic parity of $p$ be negative, since I think each of $\rho^+$ and $\pi^-$ have negative intrinsic parity?