I will not answer the question since I think
it is not completely defined without completely specifying the machine model.
In the following I explain why.

Note that the exact running time of a machine is not natural mathematically,
it heavily depends on the particular model of the machine we use
(that is why in complexity theory
we get rid of such constants using asymptotic notations like $O$).
Therefore,
the question of computing the running time modulo some value is not natural either.

We can easily consider a machine model
where all halting machines halt in even number of steps and
another model where all halting machines halt in odd number of steps,
and in both cases your question is trivially computable.
Even more artificially,
we can define a machine model where
a halting machine with size $k$ halts in even number of steps iff
a particular machine halts in $k$ steps (which would make the problem uncomputable).
These are all valid models of computation though artificial ones.

**The function is trivially computable in one model of computation,
and uncomputable in another model of computation**.
That tells us that the question is not a natural one.
Note that this is not the case regarding $BB$,
it is uncomputable in all acceptable models of computation
(having same computational power as Turing machines).
The answer doesn't depend on the particular model of computation we use,
the numbers will be different,
but the question about computability has the same answer in all of them.

Your question about $BB(k) \mod 2$ for
a particular fixed completely specified model of computation is well defined,
but IMHO it is not a very interesting one:
it is too much dependent on the particularities of the machine model.
Unless you have some reasonable restrictions on the models such that
the answer will be the same on all such models
the question would not be very interesting IMHO.

In a sense, this is similar to asking
if a particular Diophantine equation has any integer answers
without explaining why that particular equation is of any interest.
Here is the same, from mathematical perspective,
the answer to the question is not a natural one and
too much dependent on the particular encoding of the machines
and I don't see any way of getting around this dependence.
(Also from practical perspective,
I also don't see any use for computability of this function.)

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