Решение:
- His parents let him go to the party. (Глагол 'let' требует инфинитива без частицы 'to').
- They decided to sell their old car. (Глагол 'decide' требует инфинитива с частицей 'to').
- Joan spent all day shopping. (Конструкция 'spend time doing something' требует герундия).
- I'm busy doing my homework at the moment. (Конструкция 'be busy doing something' требует герундия).
- I'd love to visit India one day. (Глагол 'love' после 'would' требует инфинитива с частицей 'to').
- Penny loves visiting museums. (Глагол 'love' требует герундия).
- His teacher made him apologise for his bad behaviour. (Глагол 'make' требует инфинитива без частицы 'to').
- Jane isn't used to getting up early in the morning. (Конструкция 'be used to doing something' требует герундия).
- The boys went hiking in the woods yesterday. (Конструкция 'go + -ing' обозначает занятие).
- Nikos agreed to buy my old laptop. (Глагол 'agree' требует инфинитива с частицей 'to').
- Swimming keeps you fit. (Герундий как подлежащее предложения).
Ответ: go; to sell; shopping; doing; to visit; visiting; apologise; to getting up; hiking; to buy; Swimming.