pleasehelp“They had to have that specially made?”
The sentence is in the past tense. Since English lacks verbs for situations in which person A does something for person B on person B's request, a rather long and awkward structure is used instead: to have something done. Have is the finite verb, in other words, the verb that shows tense. The past participle has absolutely nothing to do with tense as it is always required regardless of the tense.
Other examples of the same structure in various tenses:
Present tense: He has his house painted every five years.
Past tense: He had his house painted last week.
Present perfect: He has had his house painted twice.
Past perfect/Pluperfect: He had had his house painted needlessly.
Present conditional: He would have his house painted if he had the money.
Perfect conditional: He would have had his house painted if he had had the money.
Future tense: He will have his house painted next year.
Present continuous: He is having his house painted right now.
Interrogative past tense: Did he have his house painted last year?
And so on.
English verb forms are the simplest I can think of. The morphology is extremely easy! What is difficult is that these verb forms and the defective/modal auxiliaries in particular are sometimes used very illogically.
To know how to use the above structure, you only have to know how to use have, that's all!
CB