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First, the "standard" way to do this is to change the base ring of the matrix as follows;

sage: B = A.change_ring(ZZ) ; B
[1 2 3]
[4 5 6]
sage: B.parent()
Full MatrixSpace of 2 by 3 dense matrices over Integer Ring

If P is a parent and E is some element, P(E) tries to convert E as an element of P. Hence, when you do B = ZZ(A), Sage tries to transform the matrix into an integer, which leads to an error. If you want to use such a conversion, the parent P you are looking for is the space of integer matrices of size 2*3 :

sage: P = MatrixSpace(ZZ,2,3)
sage: P
Full MatrixSpace of 2 by 3 dense matrices over Integer Ring
sage: P(A)
[1 2 3]
[4 5 6]

Or directly:

sage: MatrixSpace(ZZ,2,3)(A)
[1 2 3]
[4 5 6]

Note that it leads to the same result as with the change_ring method:

sage: P(A) == B
True

sage: P == B.parent()
True