Your puzzle isn't right about how bash (and the shell in general) parsed the input. In:
DIR="$( cd "$( dirname "${BASH_SOURCE[0]}" )" && pwd )"
First, bash parse the right hand side of assignment to one long string $( cd "$( dirname "${BASH_SOURCE[0]}" )" && pwd ) because double quote can appear inside double quotes.
After then, bash start parsing the command substitution. Because all characters following open parenthesis to enclosing parenthesis are used to construct the command inside command substitution, you will get:
cd "$( dirname "${BASH_SOURCE[0]}" )" && pwd
The shell continue parsing that compound command, break it into two parts:
cd "$( dirname "${BASH_SOURCE[0]}" )"- pwd
Then applying the same parsing rule for cd "$( dirname "${BASH_SOURCE[0]}" )", but this time, double quotes are not redundant, but make sense. They prevent field splitting on result of $( dirname "${BASH_SOURCE[0]}" ), and also the expansion of ${BASH_SOURCE[0]} (In contrast with the outer most double quotes, will are not necessary in RHS of variable assignment to prevent split+globare not necessary in RHS of variable assignment to prevent split+glob).
This rule apply to command substitution in all POSIX shell. A more details puzzle you can read in Token Recognition section of POSIX spec.
