This is not an additional answer, since I believe "carga de la prueba" is as close as you can get in Spanish to "burden of proof". But for the sake of the glossaries I wanted to underscore that "carga de la prueba is not actually the same as "burden of pursuasion." "Carga de la prueba" is actually "burden of proof" and in Anglo-American law the concept of burden of proof entails two different aspects: "burden of pursuasion" and "burden of production", also called "burden of going forward with the evidence" ("carga de aportación de la prueba").

This can be seen in the "Black's Law Dictionary" (8th ed.) definitions:

Burden of proof--A party's duty to prove a disputed assertion or charge. The burden of proof includes both the burden of persuasion and the burden of production.

Burden of persuasion--A party's duty to convince the fact-finder to view the facts in a way that favors that party. In civil cases, the plaintiff's burden is usually "by a preponderance of the evidence", while in criminal cases the prosecution's burden is "beyond a reasonable doubt."

Burden of production--A party's duty to introduce enough evidence on a issue to have the issue decided by the fact-finder, rather than decided against the party in a peremptory ruling such as a summary judgment or directed verdict.

The fact that the Spanish concept of "burden of proof" may not have exact terminological equivalents for both "burden of persuasion" and "burden of production" was underscored by Prof. Fernando Gómez Pomar (professor in the law school of the Universitat Pompeu Fabra in Barcelona) in an article entitled "Carga de la prueba y responsabilidad objetiva", published in InDret 1 (2001):

Las reglas sobre carga de la prueba comprenden, pues,
de un lado, la determinación del umbral de certidumbre que requiere el juzgador para satisfacer la pretensión y, de otro, la determinación de cuál de las partes ha de suministrar las pruebas para alcanzar dicho umbral, so pena de recibir una decisión adversa sobre el fondo del asunto si no lo hace.

En el ámbito jurídico norteamericano, ambos aspectos son analizados independientemente dentro de la genérica “burden of proof”: se habla así de “burden of persuasion”, “level of confidence” o “standard of proof” (son los términos que emplean, respectivamente, Richard POSNER, 1999, p. 1477, Bruce
HAY y Kathryn SPIER, 1997, p. 414 y Michael DAVIS, 1994, p. 346) para referirse al primero, frente a “burden of production” o “burden of proof” en sentido estricto.

En España y, en general, en Europa, no se realiza con nitidez –o no se realiza en absoluto- la distinción entre las dos vertientes de la carga de la prueba. Tal vez ello se deba a que implícitamente se considera que sólo hay un nivel de confianza o convicción jurídicamente admisible en el juzgador acerca del acaecimiento de un cierto suceso.
http://www.indret.com/pdf/040_es.pdf