The wording “expired party” is very odd in this context. I had never come across it before. I am used to seeing “unsuccessful party” in arbitration proceedings instead. Anyway, the translation I suggest for the query term is “parte vencida”.

From Mexico:
https://mexico.justia.com/federales/codigos/codigo-de-comerc...
Código de Comercio
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Artículo 1455. Salvo lo dispuesto en el párrafo siguiente, las costas del arbitraje serán a cargo de la parte vencida. Sin embargo, el tribunal arbitral podrá prorratear los elementos de estas costas entre las partes si decide que el prorrateo es razonable, teniendo en cuenta las circunstancias del caso.

As you can see, the expression “expired party” does exist (but I don´t trust this source very much, to tell the truth):
https://www.global-regulation.com/translation/mexico/560098/...
Commercial Code
Original Language Title: Código de Comercio
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Article 1455.- Except as provided in the following paragraph, the costs of the arbitration shall be borne by the expired party. However, the arbitral tribunal may prorate the elements of these costs between the parties if it decides that the pro rata is reasonable, taking into account the circumstances of the case.

And this is what I believe is trustworthy: “Unsuccessful party” (parte vencida), as you can see here, for instance:
https://uk.practicallaw.thomsonreuters.com/0-617-6111?transi...
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Article 42(1) of the UNCITRAL Arbitration Rules, which provides that the costs of the arbitration shall be borne by the unsuccessful party, was excluded from the terms of the parties' arbitration agreement. Accordingly, the claimant and respondent each bore the costs of their appointed co–arbitrator and all their own costs, and the costs of the presiding arbitrator equally, even though the claimant was unsuccessful in the arbitration.