Is there a word for someone who has others do all their work for them?

A delegator. Please find the definition and edit my answer.

taskmaster noun

: a person who assigns work to other people

: one that imposes a task or burdens another with labor

"Taskmaster" can be used gender-neutral but there is taskmistress also.

Example usages:

• They set taskmasters over us in order to oppress us with our burdens.
• The soul, the higher self, is not a hard taskmaster.
• Parents can be tough taskmasters in seeking the best for their children, as many of you may know.

Note: There is a connotation of "assigning" in this word.

After your explanations, these might fit: (Including slang words from different English speaking countries. You might check thesaurus for these words as well.)

drone: An idle person who lives off others; a loafer.

skiver: (British, slang) a person who persistently avoids work or responsibility

bludger: (Australian & New Zealand, informal)

• a person who avoids work

• a person in authority regarded as ineffectual by those working under him

goof-off: (Slang) One who shirks work or responsibility.