The questionnaire linked in your question is specifically about visiting family in the UK. Since you will be visiting for work-related purposes, that doesn't apply. The page you're looking for is this one, which says:
If you’re visiting for certain business or academic activities
You can come to the UK as a Standard Visitor for up to 6 months without a visa, but you can only do certain business or academic activities, for example go to a conference or a meeting.
- do paid or unpaid work for a UK company or as a self-employed person
- do a work placement or internship
- sell directly to the public or provide goods and services
That's still a bit vague, and they provide a link to the Immigration Rules Appendix Visitor: Permitted Activities. Under work-related training it says:
PA 10.2. Employees of an overseas company or organisation may receive training from a UK based company or organisation in work practices and techniques which are required for the visitor’s employment overseas and not available in their home country.
Based on your description it's not clear if the training you intend to follow meets those criteria. Specifically, it's not clear if the training is not available in your home country. As such, I think the following options is the most logical: Ask the UK parent company whether their training qualifies under PA 10.2 of the UK's Immigration Rules Appendix Visitor: Permitted Activities, i.e. whether you may attend the training as a visitor to the UK.
Another approach would be to get a work visa just in case. In that case, you would need to get it through your employer anyway so they can sponsor your visa. So even when you choose this option the employer (or the UK parent company) seems like the best starting point for your question.
As for the work-related training exception, there's some guidance in this document by the Home Office (specifically related to the PA 10.2 allowed activity):
Training should be in work practices and techniques that are not available in the
visitor’s home country. It should typically be class-room based or involve
familiarisation or observation. Practical training is however allowed provided it does
not amount to ‘training on the job’ or the person filling a role. It is acceptable for a
visitor to learn how to use a piece of equipment in the UK, but you must carefully
assess how long they intend to do this for and make sure there is no risk that they
will be working for that company in the UK.
Where you think the training is available in their home country, you may want to
question why the visitor needs to come to the UK.
If the visitor states that they will be being trained for longer than one month, you
should consider who will be covering their work overseas and whether their training
activities actually amount to taking employment in the UK.
Based on that, it seems the main criteria are that the activity in the UK doesn't amount to actual work and the training is not available back in the US. Again, this is something the UK parent company knows because they will most likely have dealt with foreign trainees before.
As for your wife, whether she needs a visa depends on what she intends to do during your time in the UK. If she wants to work then she'll need a visa. If the idea is that she'll just accompany you and maybe do some sightseeing then she can stay as a visitor based on the page you found already. Of course the two of you may need to convince UK officials at the border that she won't be working or doing anything else that you're not allowed to do as a visitor. For starters, that means you need to convince them that you have sufficient funds or income to support yourselves during your stay.
When you select the visiting as a tourist option, the following pops up:
What you need at the UK border
You must provide a valid passport or travel document. Your passport should be valid for the whole of your stay in the UK.
You may also be asked to prove that:
- you’re visiting for tourism
- you’re able to support yourself and your dependents during your trip (or have funding from someone else to support you)
- you’ve arranged accommodation for your stay
- you’re able to pay for your return or onward journey (or have funding from someone else)
- you’ll leave the UK at the end of your visit
I'd expect similar questions in your situation.