When DCF has custody of a child pursuant to a care and protection petition it has "the power to: (1) determine a child's place of abode, medical care and education; (2) control visits to a child; and (3) consent to enlistments, marriages and other contracts otherwise requiring parental consent." Mass. Gen. Laws chapter 119, section 21, linked below.
Are you saying that the child was placed with you as a kinship foster care provider? If so, generally, DCF gives foster parents the discretion to include the foster children in family activities UNLESS the activity would put the child in contact with a parent from whom the child was removed in violation of the parent's service plan / visitation arrangement, another person who has victimized the child or a person with certain types of CORI issues.
If your placement agreement with DCF doesn't specifically limit where you can take your foster child, the child's case manager / social worker hasn't told you otherwise and none of the people I mentioned above will be there, you are probably safe in taking the child to a cousin's birthday party.
If you have any doubts, you could talk to the child's worker or supervisor. If you don't like the answer you get you can file a grievance with director of the area office that is handling the child's case (though those take a long time and are often ignored). See the last three subsections, 10.37-10.39, of the DCF "Fair Hearings and Grievances" regulation linked below.
The statute also says, "If a parent or guardian objects to the [the way DCF is] carrying out [ ] any power conferred by this paragraph, that parent or guardian may take application to the committing court and the court shall review and make an order on the matter." So, if DCF denies you the opportunity to take the child to the party and the child's parent(s) object to it, the parent(s) can have his/her/their lawyer(s) file an abuse of discretion motion with the juvenile court judge who gave custody to DCF.
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