Learn French: Je prends la chaise de peur que le garçon grimpe. - I am taking the chair for fear that the boy might climb.

The conjunction de peur que (for fear that) expresses an emotion and a desire to avoid a potential outcome, which always triggers the subjunctive mood in French.

grimpe looks exactly like the normal present tense. How do we know it's the subjunctive?

For regular -er verbs like grimper, the present subjunctive forms for je, tu, il/elle/on, and ils/elles are spelled identically to the normal present tense. Even though the spelling doesn't change, grammatically it is acting as the subjunctive here because it directly follows de peur que.

Shouldn't there be a ne before grimpe, like de peur que le garçon ne grimpe?

In formal or written French, you will usually see an extra ne (called the 'ne explétif') after de peur que. It doesn't make the sentence negative; it just emphasizes the feeling of fear or avoidance. However, in everyday spoken French, it is very common to leave it out, exactly as this sentence does.

Can I use parce que j'ai peur que instead of de peur que?

Yes, the meaning is essentially identical. Saying parce que j'ai peur que le garçon grimpe is perfectly correct and very common in conversational French. de peur que is just a more concise, slightly more elevated way to express the exact same idea, similar to 'lest' or 'for fear that' in English.