  Parallel addition in integer base is used for speeding up multiplication and
division algorithms. $k$-block parallel addition has been introduced by
Kornerup in 1999: instead of manipulating single digits, one works with blocks
of fixed length $k$. The aim of this paper is to investigate how such notion
influences the relationship between the base and the cardinality of the
alphabet allowing parallel addition. In this paper, we mainly focus on a
certain class of real bases --- the so-called Parry numbers. We give lower
bounds on the cardinality of alphabets of non-negative integer digits allowing
block parallel addition. By considering quadratic Pisot bases, we are able to
show that these bounds cannot be improved in general and we give explicit
parallel algorithms for addition in these cases. We also consider the
$d$-bonacci base, which satisfies the equation $X^d = X^{d-1} + X^{d-2} +
\cdots + X + 1$. If in a base being a $d$-bonacci number $1$-block parallel
addition is possible on the alphabet $\mathcal{A}$, then $\#\mathcal{A} \geq
d+1$; on the other hand, there exists a $k\in\mathbb{N}$ such that $k$-block
parallel addition in this base is possible on the alphabet $\{0,1,2\}$, which
cannot be reduced. In particular, addition in the Tribonacci base is $14$-block
parallel on alphabet $\{0,1,2\}$.
