However, analysis also showed that there were 145 excess cases per 10 million people after testing positive for Covid – almost four times more than following a jab.
Data from more than 30 million people in England was used to assess how the vaccine and virus impacted on a person's chance of developing adverse neurological complications post-vaccination and post-infection.
The analysis found that a person who had the AstraZeneca jab was almost three times more likely to develop Guillain-Barré syndrome two to three weeks after their inoculation compared to if they did not get the jab.
However, the risk was significantly higher if they caught Covid, making them more than five times as likely to develop the condition within a month of testing positive.
The researchers also investigated Bell's palsy, another potentially serious side-effect which can lead to short-term weakness of the facial muscles.
Data show that, 15 to 21 days after getting the AstraZeneca jab, a person was 29 per cent more at risk of the condition than normally. In contrast, a person who caught Covid was 34 per cent more likely to develop Bell's palsy within a month of testing positive.
Professor Aziz Sheikh, professor of primary care research and development at the University of Edinburgh and a co-author of the study, said: "The risks are orders of magnitude higher if people get infected.
"But ultimately these are people's decisions and they need to make decisions for themselves. What we are trying to do is provide the best data possible and frame this so people are informed of the risks – there are risks clearly associated with the vaccine, but there are more substantial risks associated with infection.
"We are not seeing a higher risk for any of these adverse events associated with vaccines that associate with the infection."
The researchers also found there was a very small increased risk of "haemorrhagic stroke" within 28 days of getting the Pfizer vaccine, estimated as only 60 extra cases per 10 million people.
But Professor David Werring of UCL, who was not involved with the research, said this finding should be treated with caution.
"Importantly, following a positive SARS-CoV-2 test, the authors found a substantially higher risk of all the neurological diseases studied, emphasising that the benefits of ongoing vaccination efforts worldwide outweigh these potential risks."
