Bradford City underestimate ticket demand for Reading game
Season-ticket holders were allowed to purchase as many tickets as they wanted, at the club's discretion.
The club sold out an initial allocation on Sunday, leaving some season-ticket holders out in the cold without a ticket.
Rhodes told BBC Sport: "The priority is to get them a ticket. The speed at which they sold out surprised us."
The same ticketing policy was used for the Bantams' win over Sunderland in the previous round, and Rhodes said it had "worked well".
Bradford sold out for the tie against the Black Cats, which they won 2-0, resulting in a capacity crowd of 24,021.
It was was the biggest attendance for a Bantams home game since the stadium was rebuilt after the fire disaster of 1985, which claimed 56 lives.
Rhodes added: "The Sunderland game was a huge success and we wanted to recreate that atmosphere and give season-ticket holders the chance to bring a friend or a member of their family.
"We didn't want to replicate the Aston Villa game in the semi-final of the League Cup in 2013, where we didn't sell out. We have tried to balance that with satisfying season-ticket holders.
"But we don't want season-ticket holders to miss out and they are our priority. We have kept some tickets back and have advised season-ticket holders without tickets to contact the ticket office."
Some fans have bought more tickets than they needed for City's first FA Cup quarter-final since 1976 and have been attempting to sell them online at inflated prices.
"We can't stop people from selling tickets on once they've bought them but it is illegal," Rhodes's co-chairman Mark Lawn told Talksport.
"I wouldn't have expected Bradford City fans to go and make money from their fellow fans because we are not that type of club."
Lawn has threatened to ban any fans found to have been attempting to sell tickets on.
The Bantams agreed with Reading to sell all tickets at £15 for an adult, £10 for students and £5 for fans aged under 16 - the same prices they set for the fifth round win over Premier League Sunderland.
Lawn said he remained confident that all remaining season-ticket holders who have not been able to buy seats yet should all be able to do so on a one ticket per person basis.
Meanwhile, Reading have put an extra 1,587 tickets on sale after their initial allocation of just over 2,700 sold out on Monday morning.
Ironically, despite this game being live on TV unlike the Sunderland game, demand has been far higher.