Does parR imply Souslin? I have encountered the following property in this article:

We say that a space $X$ parR (partition-Rothberger), if, for every sequence $(\mathcal P_n : n \in \omega)$, of partition of $X$ into clopen sets, one can pick $V_n \in \mathcal P_n$, so that $\{ V_n : n \in \omega \}$ covers $X$.

also:

A topological space $X$ has the Souslin property if every pairwise disjoint family of non-empty open subsets of $X$ is countable.

I am trying to decide weather this two qualities are exuivalent.
My guess is that parR implies Souslin and Souslin does not implies parR.
Am I right?
If yes, amy ideas for a space which is Souslin but not ParR?
Thank you!
 A: An example of a (zero-dimensional) space with the Souslin property but not the partition-Rothberger property is the Sorgenfrey line (lower limit topology on $\mathbb{R}$).  It is separable, therefore clearly has the Souslin property.  But if for each $n \geq 0$ you take $P_n$ to be a partition of $\mathbb{R}$ into half-open intervals $[x , x+2^{-n} )$, then you cannot cover $\mathbb{R}$ by picking one $V_n$ for each $P_n$.  (A similar reasoning as the one behind showing that the real line does not satisfy the Rothberger property $\mathsf{S}_1 ( \mathcal{O} , \mathcal{O} )$.)
For the partition-Rothberger property, note that some spaces have this property for trivial reasons.  For example, every connected space $X$ has the partition-Rothberger property (since the only clopen subsets are $\varnothing$ and the entire space any partition of $X$ into clopen sets must contain $X$).  There are examples of connected spaces that do not have the Souslin property, like the long line.

Addendum
The paper being read appears to only consider the partition-Rothberger property in the context of zero-dimensional spaces (i.e., spaces with a clopen base).    There are zero-dimension spaces with the partition-Rothberger property which do not have the Souslin property.  Perhaps the simplest example is the ordinal space $\omega_1 + 1 = [ 0 , \omega_1 ]$.


*

*Note that $\{ \{ \alpha+1 \} : \alpha < \omega_1 \}$ is an uncountable family of pairwise disjoint open sets.

*For the partition-Rothberger property, suppose that $\{ \mathcal{P}_n \}_{n \in \omega}$ is a family of partitions of $[0,\omega_1]$ into clopen subsets.  First pick the unqiue $U_0 \in \mathcal{P}_0$ containing $\omega_1$. Note that $[0,\omega_1] \setminus U_0$ must be countable, so enumerate it as $\{ \alpha_n \}_{n \geq 1}$.  For each $n \geq 1$ pick $U_n \in \mathcal{P}_n$ containing $\alpha_n$.  Then $\bigcup_{n \in \omega} U_n = [0,\omega_1]$.
