The word "get" gets used in many colloquial expressions, and it's a good idea to get accustomed to avoiding such usage. Having said that, the word "get" can take both an infinitive form (to + verb) and a gerund form (verb + ing), although the meaning changes slightly between the two.

If you get somebody to do something, it usually means you've asked that person and he/she has agreed. A verb like "allow" means you give permission (it's not your request). By putting the verb in continuous form, you're basically saying that the action is continuous. Probably happening at that moment.

'To get someone to do something' means that you arrange or ask someone to do something, eg: I'll get the gardener to cut the grass next week. It can also be used when you persuade or convince someone to do something. This is very different from 'let someone do something' or 'allow someone to do something', both of which mean 'not prevent'. My parents only allow me to stay out late at weekends. My parents only let me stay out late at weekends. It's also different from 'make someone do something', which means to force someone to do something. The restaurant boss made me stay at work until I'd washed all the dishes. 'Get + object + gerund' is more unusual. It's used in situations such as: That was an interesting book. It got me thinking about all sorts of things.