Is there a word for people who use children to gain sympathy?

For example, beggars use children to get more money. People who think they can get away with things because they have children around.

The word can be an adjective or a noun.

A ______ person.
A ______.

Doesn't matter.

Just to be clear, at least in most of Europe, it is illegal to use children as pawns for begging. Passersby might be accosted by nomads or Romani people with infants in their arms and be asked for money but these persons will rarely remain on a fixed spot for a length of time.

This probably explains the absence of a specific word for an adult who panhandles/begs with children as the offenders can risk fines, imprisonment, or losing the custody of their children.

However if one is looking for an umbrella term that loosely fits the definition of an adult who exploits children for monetary gains then child trafficker could be used.

The Wikipedia page on child trafficking says [emphasis in bold mine]

Trafficking of children is a form of human trafficking and is defined by the United Nations as the "recruitment, transportation, transfer, harboring, and/or receipt" kidnapping of a child for the purpose of slavery, forced labor and exploitation. […] The objective of child trafficking is often forced child labour. Child labour refers specifically to children under a stipulated minimum age, usually 14 at the lowest, being required to work. UNICEF estimates that, in 2011, 150 million children aged 5–14 in developing countries were involved in child labour.

Child trafficking is a serious crime and an accurate accusation for the act of kidnaping and selling children but used loosely, it could refer to adults who use their offspring for forced labour, as it is abusing and trafficking the services of that child.

The word pawn is appropriate to describe the child whose status is being manipulated.

Source: Longman Dictionary

pawn

2. someone who is used by a more powerful person or group and has no control of the situation.

• The ambassador was being used as a pawn in the struggle between the two superpowers.

Source: Cambridge Dictionary

A pawn is also a person who is controlled by others and used for their own advantage:

• She felt she had been used as a political pawn.

From NBC New York

“If it’s even possible that parents are using their babies as pawns to get people to give them donations that’s horrendous and deeply unfair to those children,” de Blasio said at the time, saying he would put his inter-agency children’s cabinet and the NYPD on the case. (2015)

A adjective for taking advantage of others (including but not limited to one's children) is

exploitative (adj.)

Unfairly or cynically using another person or group for profit or advantage m-w

Making use of a situation or treating others unfairly in order to gain an advantage or benefit. Lexico

exploit (v.) To make use of meanly or unfairly for one's own advantage m-w

exploiter (n.) A person who takes advantage of something, esp. a person who takes unfair or unethical advantage of another OED

Exploitative parents are focused only on their needs. They view their children as a means of solving their problems. They wish to reverse roles with their children and to remain forever dependent on their sons and daughters. A. M. Uhl; The Complete Idiot's Guide to Coping with Difficult People

Côte d'Ivoire is one of nine countries participating in the ILO-IPEC project to combat the trafficking of children for exploitative labor in West and Central Africa... U.S. Dept. of Labor; 2001 Findings on the Worst Forms of Child Labor

Rather than the family being a protective haven, it was the place where, at best, parents might exploit their children and treat them as a mixture of expensive nuisance, slave and 'Ideal Cute Child'; at worst, a place where parents could abuse their children in private. Jane Fortin; Children's Rights and the Developing Law p.8

Young can do whatever they want. Parents support that. Many parents want to present themselves as modern and open-minded. In reality, they are disinterested and just care about their ego. They exploit their children both financially and emotionally. So these people don't only exploit their children, but also tie them to themselves. V. Živković; Return to God