How many dark red balls are there in the box?

The total number of balls is given as $1000$.

The percentage of red balls is $21\%$.

Hence, the number of red balls is calculated as:

$\text{Number of red balls} = 21\% \times 1000 = \frac{21}{100} \times 1000 = 210$

The light-to-dark ratio for red balls is $3:4$.

This means, for every $7$ parts ($3 + 4$), $3$ parts are light, and $4$ parts are dark.

The number of dark red balls is:

$\text{Dark red balls} = \frac{4}{7} \times 210 = 120$