I'm assuming they are objecting to your motion to quash on the grounds that the rule does not provide that one party may file the objection on behalf of the third party. There is no case law that I have located which specifically speaks to this point in Arizona.
Lets start with the rule, just to make sure that you are aware of it. It is Arizona Rule of Civil Procedure
45, which in part states:
(e) Protection of Persons Subject to Subpoenas; Motion to Quash or Modify.
(1) Avoiding Undue Burden or Expense; Sanctions. -- A party or an attorney responsible for the service of a subpoena shall take reasonable steps to avoid imposing undue burden or expense on a person subject to that subpoena. The issuing court shall enforce this duty and impose upon the party or attorney who breaches this duty an appropriate sanction, which may include, but is not limited to, lost earnings and a reasonable attorneys' fee.
(2) Quashing or Modifying a Subpoena.
(A) When Required. -- On the timely filing of a motion to quash or modify a subpoena, the superior court of the county in which the case is pending or from which a subpoena was issued shall quash or modify the subpoena if:
(i) it fails to allow a reasonable time for compliance;
(ii) it commands a person who is neither a party nor a party's officer to travel to a location other than the places specified in Rule 45(b)(3)(B);
(iii) it requires disclosure of privileged or other protected matter, if no exception or waiver applies; or
(iv) it subjects a person to undue burden.
(B) When Permitted. -- On the timely filing of a motion to quash or modify a subpoena, and to protect a person subject to or affected by a subpoena, the superior court of the county in which the case is pending or from which a subpoena was issued may quash or modify the subpoena if:
(i) it requires disclosing a trade secret or other confidential research, development, or commercial information;
(ii) it requires disclosing an unretained expert's opinion or information that does not describe specific occurrences in dispute and results from the expert's study that was not requested by a party;
(iii) it requires a person who is neither a party nor a party's officer to incur substantial travel expense; or
(iv) justice so requires.
(C) Specifying Conditions as an Alternative. -- In the circumstances described in Rule 45(e)(2)(B), the court may, instead of quashing or modifying a subpoena, order appearance or production under specified conditions, including any conditions and limitations set forth in Rule 26(c), as the court deems appropriate:
(i) if the party or attorney serving the subpoena shows a substantial need for the testimony
or material that cannot be otherwise met without undue hardship; and
(ii) if the person's travel expenses or the expenses resulting from the production are at issue, the party or attorney serving the subpoena assures that the subpoenaed person will be reasonably compensated.
(D) Time for Motion. -- A motion to quash or modify a subpoena must be filed before the time specified for compliance or within 14 days after the subpoena is served, whichever is earlier.
(E) Service of Motion. -- Any motion to quash or modify a subpoena shall be served on the party or the attorney serving the subpoena in accordance with Rule 5(c). The party or attorney who served the subpoena shall serve a copy of any such motion on all other parties in accordance with Rule 5(c).
The following is some case law which you might be able to use:
"A trial court may, however, make any order which justice requires to protect a party or person from annoyance, embarrassment, oppression, or undue burden or expense."
Am. Family Mut. Ins. Co. v. Grant, 222 Ariz. 507 (Ariz. Ct. App. 2009)
"In the event of a dispute between the parties, or between parties and third persons, as to a particular item's relevance, the ultimate determination is a judicial one. Rule 26(c) is a uniquely appropriate tool for the court to utilize in this regard for it permits judicial restrictions to be imposed on the discovery process and confers upon the courts considerable flexibility and discretion in selecting various means to protect a party or person from annoyance, embarrassment, oppression, or undue burden and expense. 8 C. Wright & A. Miller, Federal Practice and Procedure § 2036 (1970). See Tucson Medical Center Incorporated v. Rowles, 21 Ariz.App. 424, 520 P.2d 518 (1974). In order for the court to determine the relevancy of a specified subject matter, it may become necessary for the party claiming irrelevancy to produce the documents under seal for an in camera inspection by the court, together with supporting factual details of each item claimed to be irrelevant. See Hunter v. International Systems & Control Corp., 51 F.R.D. 251 (W.D.Mo.1970)."
Jolly v. Superior Court, 112 Ariz. 186, 192 (Ariz. 1975)
"Many jurisdictions require litigants to first pursue less intrusive discovery before resorting to broad demands for information such as the subpoena at issue here. See, e.g., Elkins, 672 So.2d at 521-22 (overturning trial court's enforcement of broad subpoena to experts, noting, "[t]he [***15] least burdensome route of discovery . . . was simply not followed."); Primm v. Isaac, 127 S.W.3d 630, 638 (Ky. 2004) ("As [the claimant] has yet to take Dr. Primm's deposition and question him about the sought-after information, the least burdensome route of discovery was simply not followed."); Cooper, 905 A.2d at 495 (holding expert bias discovery "should be of the least burdensome and intrusive kind possible"). See also State Farm, 167 Ariz. at 139, 804 P.2d at 1327 (noting less onerous methods of discovery bad faith plaintiff could employ in lieu of the sweeping requests at issue). Such a requirement has both legal and equitable merit."
Am. Family Mut. Ins. Co. v. Grant, 222 Ariz. 507, 513 (Ariz. Ct. App. 2009)
Requiring litigants to at least initially pursue less intrusive discovery before resorting to sweeping demands for information respects the competing interests outlined in P 16. It is also consistent with the mandate that the HN11rules of civil procedure, including [***17] those relating to discovery, "be construed to secure the just, speedy, and inexpensive determination of every action." Ariz. R. Civ. P. 1.
Am. Family Mut. Ins. Co. v. Grant, 222 Ariz. 507, 513 (Ariz. Ct. App. 2009)
Thus the burden to establish that a subpoena duces tecum is unreasonable or oppressive is on the party who seeks to have it quashed. He cannot rely on the mere assertion that compliance would be burdensome or onerous without showing the manner and extent of the burden and the injurious consequences of compliance. United States v. International Business Machines Corp., 83 F.R.D. 97, 104 (S.D.N.Y.1979); Ghandi v. Police Department of City of Detroit, 74 F.R.D. 115, 124 (E.D.Mich.1977); 9 Wright & Miller, Federal Practice and Procedure: Civil § 2457 at 435.
Helge v. Druke, 136 Ariz. 434, 438 (Ariz. Ct. App. 1983)
The designation of documents sought to be discovered must have sufficient particularity to enable the person who has possession
, custody and control thereof to know what is required. Kirkpatrick v. Industrial Commission, supra. A blanket request for all written statements, all memoranda and other documents in defendant's possession lacks specificity and is too sweeping and undetailed to comply with the rule requirements as to designation. Dean v. Superior Court in and for County of Maricopa, supra.
Helge v. Druke, 136 Ariz. 434, 439 (Ariz. Ct. App. 1983)