Does no parents imply variable independence in a Bayesian network?
Yes, they would be unconditionally independent.

This is simply a consequence of the factorisation of a Bayesian network into conditional probabilities. For example, this network will factorise as $p(A,B,C)=p(C|A,B)p(A)p(B)$, implying that $p(A,B)=p(A)p(B)$, which is the very definition of unconditional independence for the parent-less nodes $A$ &amp; $B$:



As for the factorisation $p(A,B,C)=p(C|A,B)p(A)p(B)$ that led us here, this is definitional for Bayesian networks. I recommend Section 3.3 of Barber for a fuller explanation.

On the other hand, they are conditionally dependent if conditioned on a collider. For example, nodes $x$ and $y$ are graphically conditionally dependent when conditioning on the collider $z$ in each of these 2 BNs:



Chapter 3 of the same source contains a fuller description of BNs and similar graphical networks, together with various possible tests for conditional independence.