I am a tutor working in Greater Manchester. I'd say parents get a tutor for the following reasons.
1/ Students have a mild to moderate specific learning difficulty (eg dyslexia) This may not have been picked up on by the school, especially if the child is quite bright and has developed "coping" mechanisms. Parents sense that something is not right and the child is frustrated but school repeatedly says there isn't a problem, so they look for a tutor. (I think this is partly a consequence of the national curriculum as if the child is meeting targets for their age, a lot of teachers assume they are OK or even worse say the child is lazy or not concentrating - also a surprising number of teachers don't know much about special needs and so don't always pick up on things). On the other hand the school may be aware of the problems and they might have been formally diagnosed but parents feel that the child is not making any/enough progress and is getting discouraged. Or they may just want to supplement the work that the school is doing. This forms about half my workload at present.
2/ Students have needed longer than they have been given at school to really grasp particular ideas and concepts. Their understanding begins to slide and they feel they are beginning to fail. The child starts to believe they can't do the subject. They get unhappy and may start to give up. This particularly seems to be an issue with maths and can be the result of poor teaching in school or the teaching may be good but the child just doesn't have enough time to consolidate within NC constraints before they are on to the next topic. By the time they "spiral" onto the same topic again, they have forgotten the things they half-grasped before and have to start again. Forms about a third of my workload currently.
3/ Preparation for a particular exam. This may be because the school doesn't cover the areas needed (eg 11+/independent school entry) or because the student (suddenly) realises with a bit of extra support they could achieve a grade higher (GCSE) or where there has been a lot of supply/teacher illness and the syllabus hasn't been covered properly (GCSE). Only one student in this category currently. It is interesting though that some of these students turn out to have problems with things like dyslexia too, that have never been spotted.
(It sounds like I'm finding dyslexia at every corner, but it's not me diagnosing it, just knowing some of the key signs and recommending the parent get further assessment!)
4/ The student is actually quite good at a subject but gets anxious easily and starts getting upset/worrying they can't do it. This in turn can prevent them from learning as well as they might, so then they get a lose more confidence. Tutoring can provide extra support to help the student realise s/he is capable and can do it! One student in this category at present.
5/ Specific medical reason for lack of progress such as an acquired head injury. No students in this category at present.
In general I'd say parents often get a tutor because they sense/know their child is unhappy or stressed about an area of learning. They try and address this with the school with limited/no success. They don't want their child to be unhappy so seek help. A lot of my work is about being sensitive to the emotional needs of the children as well as the academic needs.
There is an element of wanting children to do the best they can/have the best educational experience they can which sometimes translates into wanting a child to get into a particular type of school, but I've not met any truly "hothousing" type parent yet!! If parents ask me to prepare a child for 11+ or suchlike and I think they are too far off from this I am upfront and say so. Most parents know their children pretty well in any case and truly have their best interests at heart.
Not all my student are from middle class backgrounds although the majority are (as you would expect from a financial point of view). Some parents aren't well off and only afford tutoring with the help of other family members!