constructor(props){
    super(props);

    this.handleClick = this.handleClick.bind(this);
}

handleClick(number){
    console.log("NUMERO " + number);
}

<ul>
    {
        Object.keys(areas[item].childs).map(function(itemchild, ichild){
            return <li key={ ichild }>{ areas[item].childs[itemchild].description } <InputCounter handleClick={(number) => this.handleClick(number)} { ...areas[item].childs[itemchild] }/></li>
        })
    }
</ul>

increment = () => {
        if(this.state.count < this.state.max_score){
            this.setState({
                count: this.state.count + 1
            });
            //Tentei passar o novo valor
            this.props.handleClick(this.state.count);
        }
    }

render() {

        return (

            <span className="input_spinner">
              <button onClick={ this.decrement } className={`minus ${ this.state.count === 0 ? "_disabled" : ""}`}></button>
              <input  type="text" placeholder={ this.state.count } disabled/>
              <button onClick={ this.increment } className={`plus ${ this.state.count === this.state.max_score ? "_disabled" : "" }`}></button>
            </span>
        );
    }