Either term is technically correct, but I greatly prefer "drive-through" because "drive-thru" is a vulgar abbreviation (a bit like text-speak or writing "XMAS" when one means Christmas) and does nothing to enhance the English language.
It has often been said that Longleat (which started as The Lions of Longleat in 1966, before adding other animals - generally African in origin - to the mix) was the first drive-round animal reserve. This is incorrect. It was the first drive-through animal reserve OUTSIDE Africa. When Longleat proved to be exceedingly popular with the public, it was widely prophesized that safari parks would soon replace conventional zoos. This inaccurate forecast had one very big, but often overlooked, positive effect: zoo managements, scared that their visitors would desert them in favour of the new safari parks that were opening up all over the place, were forced to rebuild to give their own animals much more space in order to compete with the safari parks, and the terrible, heavily-barred, cell-like cages of the sixties began to disappear. Now that visitors had experienced seeing big animals in wide open spaces, the genie was out of the bottle, and small cages in zoos were seen as unacceptable.
In zoological circles, the founding father of the safari park, Jimmy Chipperfield, is regarded as something of a pariah, because he was essentially a businessman who was seen as exploiting animals for commercial purposes. But I hold a different view: it was, in no small part, because of him and the safari park concept that he was responsible for, that zoos suddenly reinvented themselves. Ultimately, the dire forecast that zoos had had their day was about as accurate as predictions that the world would end in 1999. In the UK the safari park trend was very short, peaking in the early 1970s. Several have long since closed down, including Windsor Safari park, Lambton Lion Park, Stapleford Lion Reserve, and Loch Lomond Bear Park. Excluding the Highland Wildlife Park, which has a drive-round component, and the drive-through reserve at Whipsnade, the youngest extant safari park in the UK (West Midland) is now nearly 40 years old. Even Blair Drummond, whilst still a safari park, is starting to drift away from the concept, by having more of its exhibits viewable by visitors on foot.