I thought this meant "por delegación", and of course "P.D." can mean that. But after looking into it I'm virtually certain that in this case, in Mexico, it doesn't.

What you have is P.D. followed by a name, and the person is described as Oficial del Registro Civil. Well, it's not impossible that this is somebody else signing on behalf of the oficial in question. But an "oficial" is not really a very important person, and it seems strange that he/she would be delegating signing rights. When you look into it, you find that names of "oficiales del registro civil" are regularly preceded by P.D. in contexts where there can be no question of signing on someone else's behalf — where it clearly doesn't mean "por delegación".

For example, here is a list of Registro Civil staff in Ixtlahuaca, with photos. They are a mixture of "auxiliares administrativos municipales" and "oficiales del registro civil". The former appear just with their name (e.g. Carlos Adán Martínez García). The latter, highlighted in green, have their name preceded by "Lic.", "P.D.", or sometimes just "C." (for ciudadano) or some other initials. Well, the ones with "Lic." are obviously Licenciados. The ones with "P.D." are "Pasantes de Derecho". Nobody is signing anything here so there's no question of "por delegación":
"LIC. JOSE ANTONIO MONDRAGON RIVERA
P.D. FILIBERTO ALANUZA ZARATE", etc.
http://dgregistro_civil.edomex.gob.mx/sites/dgregistro_civil...

Same here:
"El presente trabajo es una exposición de las experiencias adquiridas por la suscrita P.D. Candy Minelia Flores Aguayo, durante el tiempo que realicé la Práctica Profesional en la Secretaría de Relaciones Exteriores, Delegación Yucatán"
http://www.derecho.uady.mx/pictures/files/MODELO MEMORIAS PR...

The fondness of Mexicans for preceding their name with their title, usually a professional title, is well known. This is another example. Here are a couple of examples the show what it stands for:

"los miembros del Comité de Información integrado por la Pasante de Derecho Araceli Carina Jiménez Peralta, en su carácter de Presidenta Suplente, Licenciado Daniel Alcántara Cante, en su carácter de Secretario Técnico Suplente [...]"
And at the foot of this document, under Araceli's signature, her name appears as:
"PD. Araceli Carina Jiménez Peralta"
Whereas her colleagues names are preceded by "Lic.".
http://corett.net/web/images/Trnsparencia/Documentos/nuevo/8...

And another:
"DEFENSORES: LIC. JOSE LUIS CETZ HERRERA Y P.D. PATRICIA ELENA TEJERA ALVAREZ"
Toca relativo al recurso de apelación interpuesto por los defensores particulares del acusado RENALDO ALBERTO TEJERA ALVAREZ (O) REYNALDO ALBERTO TEJERA ALVAREZ, que lo son el Licenciado
JOSE LUIS CETZ HERRERA Y la Pasante de Derecho PATRICIA ELENA TEJERA ALVAREZ"
http://www.yucatan.gob.mx/docs/diario_oficial/diarios/2004/2...

So what does "pasante de derecho" mean? It means someone who has passed all the courses for the law degree but has not taken the dissertation required for final professional graduation as a licenciado. So in university contexts it means a degree candidate. That's why people put P.D. before their names on their licenciatura thesis in law, as here, for example:
https://ri.uaemex.mx/bitstream/handle/20.500.11799/62626/ARR...

Here's a definition:

"pasante
1 s m y f Persona que ha cursado y aprobado todas las materias requeridas para una licenciatura o un posgrado, pero no ha presentado todavía su examen profesional o de grado: un pasante de derecho, pasantes de doctorado"
http://dem.colmex.mx/moduls/BuscadorAvanzado.aspx

So I think you could put "[name and surnames], graduate in law (all but thesis)" or "graduate in law (without professional title)". It may be best to explain it in a note. But that, almost certainly, is what P.D. means in your context. Strictly n Mexico.

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Note added at 2 hrs (2017-02-13 23:10:17 GMT)
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There have been previous questions on P.D. in similar contexts, applying to civil registry "oficiales" in Mexico, and people have just assumed it meant "por delegación" (understandably, since outside Mexico it usually does), without realising that "P.D." before a name in Mexico has this other meaning. That's why I've gone into it at such length.

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Note added at 3 hrs (2017-02-13 23:18:05 GMT)
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As I say, it doesn't make any sense to me for an "oficial" to be delegating signing rights. There are always several "oficiales" in civil registry offices. Any of them can sign the certificates. If one isn't there, another one can do it. Why would they be delegating?

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Note added at 3 hrs (2017-02-13 23:22:32 GMT)
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I didn't explain that in Mexico, as in other Latin American countries, there is a distinction between successfully completing the course of study and obtaining the "título profesional", the "licenciatura". The former makes you an "egresado" or "pasante". The latter requires an additional undergraduate thesis and/or period of professional practice. The latter, by the way, is the reason why you'll often find "pasante" translated as "intern". There are many previous questions on this, and there's a basic explanation here:
https://en.wikipedia.org/wiki/Licentiate_(degree)#Mexico