Short answer. Under this article, yes. Article 290 says in terms that the provision does not apply to parents, guardians or persons entrusted with the custody of minors, as regards the papers or letters of the children or minors placed in their care. The exclusion is tied to that relationship.
What the law says
This provision shall not be applicable to parents, guardians, or person entrusted with the custody of minors with respect to the papers or letters of the children or minors placed under their care or custody
Revised Penal Code, Article 290 — Seizing Correspondence To Discover Secrets. Read the full provision →
The rule and its stated exception
Article 290 punishes a private individual who seizes another person's papers or letters in order to discover his secrets, and punishes it more heavily where the contents are then revealed. It also states its own limit: this provision shall not be applicable to parents, guardians, or person entrusted with the custody of minors with respect to the papers or letters of the children or minors placed under their care or custody. So the exception is not something read into the article by interpretation. It is written into the same sentence as the offence, and it addresses exactly the situation the question describes.
Custody and minority are what carry it
The exemption is framed around a relationship of care, and the categories are specific: parents, guardians, and persons entrusted with the custody of minors. The correspondence covered is that of the children or minors placed under that care or custody. An adult son or daughter still living in the family home is not a minor placed under anyone's custody, and the exemption does not reach a relative or household member who happens to be present but has no custodial role. The relationship, not the address, is what the article keys on.
It answers this article only
The question says letters, and the article says papers and letters. Messages held on a phone, a laptop or an online account are a different subject, governed by legislation passed long after the Code, with its own definitions and its own exceptions. A parent who is clear of this article is not therefore clear of everything, and the position of a school, an employer or anyone else handling a young person's communications is not addressed here at all. Where the concern is genuine safety, the practical route is documentation and advice rather than a reading of one provision.
Two questions worth keeping apart
There is what the criminal law prohibits, and there is what will actually help a family. The article settles the first, and it settles it in the parent's favour. The second is not a legal question and this page cannot answer it. If reading a child's correspondence has already produced something serious — evidence of harm, of exploitation, of an adult who should not be in contact — the useful next step is to preserve it as it stands, note the dates, and take it to someone who can act on it.