Diplomatic missions and international organisations ran up nearly £500,000 in unpaid parking fines in London last year, it was also revealed.
Diplomats and some embassy staff are entitled to diplomatic immunity.
This means they can be exempt from being tried for crimes.
The allegations, contained a written ministerial statement by new Foreign Secretary Boris Johnson, include someone at the Mexican embassy allegedly causing a child aged 13 to 15 to watch or look at an image of sexual activity.
Motoring offencesSomeone at the same embassy is accused of making an indecent image of a child and using threatening or abusive or insulting words or behaviour to cause harassment or alarm or distress.
The statement also says a member of the Saudi Arabian embassy is alleged to have committed the offence of human trafficking into the UK for the purposes of exploitation, specifically domestic servitude.
Someone at the same embassy is said to have committed the offence of human trafficking for slavery or servitude or forced or compulsory labour.
Someone at the Gabonese embassy was accused of actual bodily harm.
Staff at the embassies of the United States, China and Kazakhstan have all been accused of driving under the influence of alcohol. There is also a string of alleged motoring offences.
The alleged offences were brought to the attention of the Foreign Office by the Metropolitan Police's Parliamentary and Diplomatic Protection Command, or other law enforcement agencies.
In a written statement, Mr Johnson said: "Around 22,500 people are entitled to diplomatic immunity in the United Kingdom, and the majority of diplomats abide by UK law.
"The number of alleged serious crimes committed by members of the diplomatic community in the UK is proportionately low."
Parking finesHe said people entitled to diplomatic immunity were expected to obey the law, under the terms of the Vienna Convention on Diplomatic Relations 1961.
He added: "We take all allegations of illegal activity seriously.
"When instances of alleged criminal conduct are brought to our attention by the police, we ask the relevant foreign government to waive diplomatic immunity where appropriate.
"For the most serious offences, and when a relevant waiver has not been granted, we seek the immediate withdrawal of the diplomat."
People with diplomatic immunity had racked up 4,858 parking fines, totalling £477,499, the Foreign Office said in its statement.
Following a series of meetings with missions, subsequent payments and waived fines had knocked £161,328 off the bill, leaving £316,171 in unpaid fines.
The worst offender was the embassy of the Republic of South Sudan, which owed £53,708, followed by the High Commission for the Federal Republic of Nigeria, which owed £41,531, and the Royal Embassy of Saudi Arabia, which had a debt of £20,692.
The High Commission for the Republic of Zambia owed £18,212, the embassy of the Republic of Liberia owed £13,795, and the embassy of the Republic of the Sudan owed £13,429.
And Transport for London was owed £95,811,650 in unpaid Congestion Charge fees, with America alone owing £10,626,970 racked up from 89,308 unpaid fines
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