The right to read a Catholic Church's sacramental register?

They are meant to be secret.

1983 Code of Canon Law:

Can. 535 §1. Each parish is to have parochial registers, that is, those of baptisms, marriages, deaths, and others as prescribed by the conference of bishops or the diocesan bishop. […]

[…]

§4. In each parish there is to be a storage area, or archive, in which the parochial registers are protected along with letters of bishops and other documents which are to be preserved for reason of necessity or advantage. The pastor is to take care that all of these things, which are to be inspected by the diocesan bishop or his delegate at the time of visitation or at some other opportune time, do not come into the hands of outsiders.

1917 Code of Canon Law canon 470 is very similar, though it mentions in addition that "§ 1. The pastor […] shall also take care as far as possible to produce a book on the status of souls (status animarum)".

The right to read a Catholic Church's sacramental register?

The simple answer is that only the pastor or his delegate can claim a right to direct access to the registers. The local Ordinary (bishop) or his delegate may also have access to such records. The reason for so few persons to view these records is that these records are not so much secret, but rather private and they contain confidential information. Neither the pastor nor his delegate may disclose any confidential information!

Confidentiality

Care must be taken to protect people’s privacy. Although sacramental registers contain information about public events and other facts readily known to any interested party, they also contain information which is very personal and confidential.

The sacramental registers belong to the individual parish. They are maintained for the good of the Christian faithful; but they are private documents, not public ones. No one other than the pastor or his delegate can claim a right to direct access to the registers or to any versions or copies of them in any format.

The pastor is always and ultimately responsible for the care and confidentiality of the sacramental registers themselves as well as any reproductions. He may designate other persons to make entries in the registers and to prepare certificates. These may be employees or volunteers, but their number should be small. These designated persons must be known well to the pastor, must be capable of careful work and protecting confidentiality, and must be adequately trained to work with the registers. Their work with the registers is not to exceed their mandate from the pastor.

Sacramental registers should never be made available to genealogical researchers. The pastor or other regularly designated person may research information as requested and as time permits and make the information available on separate paper. Care must be taken not to disclose confidential information. - Handbook for Sacramental Records

This is alluded in Canon Law.

Can. 535 §1. Each parish is to have parochial registers, that is, those of baptisms, marriages, deaths, and others as prescribed by the conference of bishops or the diocesan bishop. The pastor is to see to it that these registers are accurately inscribed and carefully preserved.

[§2...]

[§3...]

§4. In each parish there is to be a storage area, or archive, in which the parochial registers are protected along with letters of bishops and other documents which are to be preserved for reason of necessity or advantage. The pastor is to take care that all of these things, which are to be inspected by the diocesan bishop or his delegate at the time of visitation or at some other opportune time, do not come into the hands of outsiders.

§5. Older parochial registers are also to be carefully protected according to the prescripts of particular law.

Some may find the following of some interest:

• Why are the Vatican archives kept secret?