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Two days after gunmen killed 12 people at French satirical magazine Charlie Hebdo in Paris, Parker said: "We know that terrorists based in Syria harbour the same ambitions towards the UK."
Security at British ports and the Eurostar terminal has been stepped up and MI5 officers have increased surveillance of British fanatics who they fear might launch copycat attacks, reports the Daily Telegraph.
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Speaking to a small audience at Thames House, the headquarters of MI5, Parker said there had been "more than 20 terrorist plots either directed or provoked by extremist groups in Syria" against western targets in the last 14 months.
Three UK terror plots have been foiled in recent months alone, said Parker, warning that "deaths would certainly have resulted otherwise".
Parker said the number of random "crude and potentially deadly" plots from "lone wolf" extremists was increasing, while it appeared that the al-Qaeda network was also still pursuing "large-scale loss of life" atrocities, such as an attack on transport systems or iconic targets.
"Although we and our partners try our utmost, we know we cannot hope to stop everything," said Parker.
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The Financial Times has described his comments as "one of the starkest assessments from any western counter-terror chief since the Syrian civil war broke out".
The MI5 chief used his speech to call for better access to digital communications, fearing that a lack of cooperation from internet companies would lead to terrorists slipping through the net.
Yesterday, France held a national day of mourning for the 12 people shot in the Charlie Hebdo attack. A minute's silence was observed and the lights on the Eiffel Tower were later turned off as a mark of respect.
Tens of thousands of troops have joined police in the hunt for the two suspects, brothers Said Kouachi and Cherif Kouachi, in a rural region of northern France.
