His account balance is considered personal financial information and falls under the same privacy protections that cover everything else about an inmate's record. The facility is not required to share it with anyone outside, including parents, and most of the time they will not.
That said, calling the counselor is your best shot at getting any information at all, and how you approach that call matters more than most people realize. Come in as a concerned, cooperative parent who is trying to make sure their son has what he needs, not as someone demanding access to information. Counselors have discretion, and a parent who is clearly invested in their child's wellbeing rather than just being nosy is more likely to get a thoughtful response. They may confirm whether there is money on the account without giving you an exact figure, or they may point you to the right department to contact.
The other option is simply to ask your son directly. If he is able to make calls or receive mail, ask him what his balance looks like and whether he needs anything. Most inmates know what is on their books and will tell a parent who is asking out of genuine concern.
If you have been sending money through a platform like JPay, TouchPay, or whatever service Tarrant County uses, you may be able to log into your own account on that platform and see the transaction history for deposits you have made, even if you cannot see the current balance on his end.
The most reliable information is always going to come from him directly.