Why should I change the subject in this situation? (V的V, V的V sentence structure)
The only difference I can see is that &quot;我的爸爸&quot; is singular, while &quot;公司的人&quot; is plural.

That is the reason.
[V 的 V] = [the one who v, does v] e.g. &quot;敵兵逃的逃，降的降&quot; = &quot;the enemy soldiers who run, do run; the enemy soldiers who surrender, do surrender&quot;. Which means 'some soldiers run, some soldiers surrender'.
打字的打字，打电话的打电话 = the ones who type are typing; the ones who make phone calls are making phone calls = 'some people are typing, some people are making phone calls'

[the enemy soldiers who run] is the subject; The second [run] is the verb

[打电话的(人)] is the subject; the second [打电话] is the verb


The reason for the second '打字' in '打字的打字' is '打字的' is a noun that means 'the one who type'. It doesn't mean he is currently typing until you add the verb '打字' after the noun.
Please see this similar question Grammar: verb + 的 + verb