It appears that the statute does allow the Association to have some restrictions on the rain barrel. I also expect your governing documents require prior approval for exterior changes. Therefore, the Association is simply following their procedures for you to obtain approval.
If the Association denies your request for a reason not mentioned in the statute, that would be a violation of the statute.
Per that
section of the statute (the applicable part):
Sec. 202.007. CERTAIN RESTRICTIVE COVENANTS PROHIBITED. (a) A property owners' association may not include or enforce a provision in a dedicatory instrument that prohibits or restricts a property owner from:
(2) installing rain barrels or a rainwater harvesting system;
(b) A provision that violates Subsection (a) is void.
(d) This section does not:
(6) require a property owners' association to permit a rain barrel or rainwater harvesting system to be installed in or on property if:
(A) the property is:
(i) owned by the property owners' association;
(ii) owned in common by the members of the property owners' association; or
(iii) located between the front of the property owner's home and an adjoining or adjacent street; or
(B) the barrel or system:
(i) is of a color other than a color consistent with the color scheme of the property owner's home; or
(ii) displays any language or other content that is not typically displayed by such a barrel or system as it is manufactured;
(7) restrict a property owners' association from regulating the size, type, and shielding of, and the materials used in the construction of, a rain barrel, rainwater harvesting device, or other appurtenance that is located on the side of a house or at any other location that is visible from a street, another lot, or a common area if:
(A) the restriction does not prohibit the economic installation of the device or appurtenance on the property owner's property; and
(B) there is a reasonably sufficient area on the property owner's property in which to install the device or appurtenance; or
(8) prohibit a property owners' association from requiring an owner to submit a detailed description or a plan for the installation of drought-resistant landscaping or water-conserving natural turf for review and approval by the property owners' association to ensure, to the extent practicable, maximum aesthetic compatibility with other landscaping in the subdivision.
(d-1) A property owners' association may not unreasonably deny or withhold approval of a proposed installation of drought-resistant landscaping or water-conserving natural turf under Subsection (d)(8) or unreasonably determine that the proposed installation is aesthetically incompatible with other landscaping in the subdivision.
Of course, you could always pay an attorney for a legal opinion.