We are tasked with determining the number of $2021$-tuples of positive integers $(a_1, a_2, \ldots, a_{2021})$ such that the number $3$ is an element of the tuple, and for each pair of consecutive elements $(a_i, a_{i+1})$, the condition $|a_{i+1} - a_i| \leq 1$ holds.

Step 1: Counting the Total Number of Tuples

First, we consider the number of possible $2021$-tuples where $a_i \in \{1, 2, 3\}$. Since each element can independently take any of the three values, there are:
$$3^{2021}$$
such tuples.

Step 2: Excluding Tuples Where 3 is Not Present

Next, we calculate the number of tuples where the number 3 does not appear. In this case, each element $a_i$ can only be either 1 or 2. Thus, there are:
$$2^{2021}$$
tuples where the number 3 is absent.

Step 3: Subtraction to Find the Desired Tuple Count

The desired number of tuples, i.e., tuples where at least one element is 3, is the total number of tuples minus the number of tuples with no '3'. Therefore, the result is given by:
$$3^{2021} - 2^{2021}$$

Hence, the number of $2021$-tuples satisfying the given conditions is:
$$\boxed{3^{2021} - 2^{2021}}$$