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I’m writing a story, but I need a word to describe somone in it who cares more about what their kid does than the child themself. Sorry if I messed up any grammar within that question.

Just to clarify, the 1st person cares more about the other’s actions than the other’s well being.

Once again I apologize for any incorrect grammar.

Robot. 'He treated his child unfeelingly, as if it were a robot, there merely to do his bidding, only criticising when things were wrong, caring not at all for the child's own feelings or needs'

Or Slave. A person with no rights who is owned by another.

'He treated the child as his personal slave...'

Automaton
Doll
Puppet
Made of wood
Heart of stone
Painted doll
Jemima (a kind of doll, the name also references slavery which might be appropriate).
Jemima doll - these originally are black dolls based on 'Aunt Jemima' a slave woman archetype

Pinocchio
The Tin Man

Pinocchio and The Tin Man are stories in which the feeling is missing - but later acquired, that you could possibly reference or plunder from.

😊

First that came to mind:

Externalizer

a person who draws a locus of control from the external world, depending on others as a source of values, ideas, and security.

An example of externalization:

"The patient is extremely argumentative; instead of recognizing this characteristic in herself, however, she complains about the difficult personalities of others and views herself as blameless"

Just to clarify, the 1st person cares more about the other’s actions than the other’s well being.

Where I'm from (southern American English), we often call such a person a user: someone who maintains a relationship solely for the purpose of exploitation or parasitism.

But it's common enough usage to be included in Urban Dictionary, where it is adequately defined.

Two synonyms of user: gold-digger, pimp

However, user may be applied to relationships such as friendships, etc., that do not involve romantic or physical activity. They're habitually parasitic and exploitative, they'll use anyone for any purpose, if they can get away with it.

• They're users; they enjoy bragging about their son's achievements, yet constantly criticize and never give him any praise or reward for his efforts.


• She's a user; she treats her daughter like a servant, while refusing her even a small allowance or some time to herself.


• He uses his son's celebrity to make himself feel famous; he likes the attention.

I can think of wicked stepmother and pushy parent. This last is less an archetype than it is a literal description of something commonly encountered. It isn't a reference to any one character, it is just a very common phrase. You didn't specify parents...

This is a loaded word in many ways, so I hesitate to use this, but have you thought about narcissist?

A person who has an excessive interest in or admiration of themselves.

In this case, they care only how the child's actions reflect upon themselves, and not about the actual child. A narcissist wants to be seen as raising a good child, but doesn't actually care whether or not they do.

Such a person who is worried over their child's actions (apparently the ones they disapprove of) even at the cost of the child's well-being is usually called a tyrant (sometimes suffixed with of a father/mother).

ODO:

tyrant
NOUN

1.1 A person exercising power or control in a cruel, unreasonable, or arbitrary way.

‘There's this tyrant of a father - whom we've changed to a tyrant of a
mother - who's saying marry this guy or die.’

‘Svensson takes on the persona of Strindberg and talks about his life
from a childhood intimidated by a tyrant of a father to the time of
his relationships with his three wives.’

‘After all, she and her husband were not tyrants.’